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CISM COURSES AND LECTURES
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The series presents lecture notes, monographs, edited works and proceedings in thefieldof Mechanics, Engineering, Computer Science and Applied Mathematics. Purpose of the series is to make known in the international scientific and technical conmiunity results obtained in some of the activities organized by CISM, the International Centre for Mechanical Sciences.
INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES COURSES AND LECTURES - No. 499
DYNAMIC METHODS FOR DAMAGE DETECTION IN STRUCTURES
EDITED BY ANTONINO MORASSI UNIVERSITY OF UDINE, ITALY FABRIZIO VESTRONI UNIVERSITY OF ROMA LA SAPIENZA, ITALY
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ISBN 978-3-211-78776-2 SpringerWienNewYork
PREFACE
Non-destructive testing aimed at monitoring, structural identification and diagnostics is of strategic importance in many branches of civil and mechanical engineering. This type of tests is widely practiced and directly affects topical issues regarding the design of new buildings and the repair and monitoring of existing ones. The load-bearing capacity of a structure can now be evaluated using wellestablished mechanical modelling methods aided by computing facilities of great capability. However, to ensure reliable results, models must be calibrated with accurate information on the characteristics of materials and structural components. To this end, non-destructive techniques are a useful tool from several points of view. Particularly, by measuring structural response, they provide guidance on the validation of structural descriptions or of the mathematical models of material behaviour. Diagnostic engineering is a crucial area for the application of non-destructive testing methods. Repeated tests over time can indicate the emergence of possible damage occurring during the structure's lifetime and provide quantitative estimates of the level of residual safety. Of the many non-destructive testing techniques now available, dynamic methods enjoy growing focus among the engineering community. Conventional diagnostic methods, such as those based on visual inspection, thermal or ultrasonic analysis, are local by nature. To be effective these require direct accessibility of the region to be inspected and a good preliminary knowledge of the position of the defective area. Techniques based on the study of the dynamic response of the structure or wave propagation, on the contrary, are a potentially effective diagnostic tool. These can operate on a global scale and do not require a priori information on the damaged area. Recent technological progress has generated extremely accurate and reliable experimental methods, enabling a good estimate of changes in the dynamic behaviour of a structural system caused by possible damage. Although experimental techniques are now well-established, the interpretation of measurements still lags somewhat behind. This particularly concerns identification and structural diagnostics due to their nature of inverse problems. Indeed, in these applications one wishes to determine some mechanical properties of a system on the basis of measurements of its response, in part exchanging the role of the unknowns and data compared to the direct problems of structural analysis. Hence, concerns typical of inverse problems arise, such as high nonlinearity, non-uniqueness or non-continuous dependence of the solution on the data. When identification techniques are applied to the study of real-world structures.
additional obstacles arise given the complexity of structural modelling, the inaccuracy of the analytical models used to interpret experiments, measurement errors and incomplete field data. Furthermore, the results of the theoretical mathematical formulation of problems of identification and diagnostics, given the present state-of-knowledge in the field, focus on quality, while practical needs often require more specific and quantitative estimates of quantities to be identified. To overcome these obstacles, standard procedures often do not suffice and an individual approach must be applied to tackle the intrinsic nature of the problem, using specific experimental, theoretical and numerical methods. It is for these reasons that use of damage identification techniques still involves delicate issues that are only now being clarified in international scientific literature. The CISM Course ^^Dynamic Methods for Damage Detection in Structures'^ was an opportunity to present an updated state-of-the-art overview. The aim was to tackle both theoretical and experimental aspects of dynamic non-destructive methods, with special emphasis on advanced research in the field today. The opening chapter by Vestroni and Pau describes basic concepts for the dynamic characterization of discrete vibrating systems. Chapter 2, by Friswell, gives an overview of the use of inverse methods in damage detection and location, using measured vibration data. Regularisation techniques to reduce illconditioning effects are presented and problems discussed relating to the inverse approach to structural health monitoring, such as modelling errors, environmental effects, damage models and sensor validation. Chapter 3, by Betti, presents a methodology to identify mass, stiffness and damping coefficients of a discrete vibrating system based on the measurement of input/output time histories. Using this approach, structural damage can be assessed by comparing the undamaged and damaged estimates of the physical parameters. Cases of partial/limited instrumentation and the effect of model reduction are also discussed. Chapter 4, by Vestroni, deals with the analysis of structural identification techniques based on parametric models. A numerical code, that implements a variational procedure for the identification of linear finite element models based on modal quantities, is presented and applied for modal updating and damage detection purposes. Pseudoexperimental and experimental cases are solved. Ill-conditioning and other peculiarities of the method are also investigated. Chapter 5, by Vestroni, deals with damage detection in beam structures via natural frequency measurements. Cases of single, multiple and interacting cracks are considered in detail. Attention is particularly focussed on the consequences that certain peculiarities, such as the limited number of unknowns (e.g., locations and stiffness reduction of damaged sections), have on the inverse problem solution. The analysis of damage identification in vibrating beams is continued in Chapter 6 by Morassi. Damage analysis
is formulated as a reconstruction problem and it is shown that frequency shifts caused by damage contain information on certain Fourier coefficients of the unknown stiffness variation. The rest of the chapter is devoted to the identification of localized damage in beams from a minimal set of natural frequency measurements. Closed form solutions for certain crack identification problems in vibrating rods and beams are presented. Applications based on changes in the nodes of the mode shapes and on antiresonant data are also discussed. Chapter 7, by Testa, is on the localization of concentrated damage in beam structures based on frequency changes caused by the damage. A second application deals with a crack closure that may develop in fatigue and the potential impact on damage detection. Chapter 8 proposes a paper by Cawley on the use of guided waves for long-range inspection and the integrity assessment of pipes. The aim is to determine the reflection coefficients from cracks and notches of varying depth, circumferential and axial extent when the fundamental torsional mode is travelling in the pipe. Chapter 9, by Vestroni and Vidoli, discusses a technique to enhance sensitivity of the dynamic response to local variations of the mechanical characteristics of a vibrating system based on coupling with an auxiliary system. An application to a beam-like structure coupled to a network of piezoelectric patches is discussed in detail to illustrate the approach. Antonino Morassi Fabrizio Vestroni
CONTENTS
Elements of Experimental Modal Analysis by F. Vestroni and A. Pau
1
Damage Identification using Inverse Methods by MJ. Friswell
13
Time-Domain Identification of Structural Systems from Input-Output Measurements by R, Beta
67
Structural Identification, Parametric Models and Idefem Code by F. Vestroni
95
Structural Identification and Damage Detection by F. Vestroni
Ill
Damage Identification in Vibrating Beams by A. Morassi
137
Characteristics and Detection of Damage and Fatigue Cracks by R. Testa
183
The Reflection of the Fundamental Torsional Mode from Cracks and Notches in Pipes by A, Demma, P. Cawley, M. Lowe and A.G. Roosenbrand 195 Damage Detection with Auxiliary Subsystems by F. Vestroni and S. Vidoli
211
Elements of Experimental Modal Analysis Fabrizio Vestroni and Annamaria Pau Dipartimento di Ingegneria Strutturale e Geotecnica, University of Roma La Sapienza, Italy Abstract Fundamental concepts for the characterization of the dynamical response of SDOF and NDOF systems are provided. A description is given of the main techniques to represent the response in the frequency domain and its experimental characterization. Two classical procedures of modal parameter identification are outlined and selected numerical and experimental examples are reported.
1 Dynamic Characterization of a SDOF The experimental study of a structure provides an insight into the real behavior of the system. In particular, the study of its dynamic response, exploiting vibration phenomena, aims to determine the dynamic properties closely connected to the geometrical and mechanical characteristics of the system. Hence, some concepts of structural dynamics will be briefly summarized. It is assumed that the reader has had some exposure to the matter (Craig, 1981; Meirovitch, 1997; Ewins, 2000; Braun et al., 2001). The classical model of a single degree-of-freedom (SDOF) system is the spring-massdashpot model of Figure 1, where the equation of motion and the steady-state solution is reported. Assuming a harmonic excitation, the frequency response function (FRF) H((jj) can be defined as the ratio between the amplitude of the steady-state response and the load intensity. The FRF shows that in a small range of the ratio UJ/UQ, when the frequency of the excitation approaches the natural frequency of the system, the response amplitude is much larger than the static response. This is called resonance. Furthermore, the amplitude of the steady-state response is linearly dependent on bothpo and H{u;). By knowing H{uj) the response of a SDOF system to a harmonic excitation can be estimated. In the real world, forces are not simply harmonic, being frequently periodic or approximated closely by periodic forces. A periodic function p{t) having period Ti can be represented as a series of harmonic components by means of its Fourier series expansion. As an example, in Figure 2, the Fourier series expansion is applied to a square wave. The Fourier series is convergent, i.e. the more terms used, the better the approximation obtained. Since the response of a SDOF system to a harmonic force is known and a periodic forcing function p{t) can be represented as a sum of harmonic forces, the response of the system u{t) to a periodic excitation can be obtained by exploiting the principle of effect superposition:
2
F. Vestroni and A. Pau harmonic excitation P i't) = Po cos {ooit) or p {t) = po sin (uJit)
equation of motion mu -i- cu-^ku
= Po sin {uit)
steady state response '^ (0 = Po-ff (k around ujr can be defined: («) {ujk) = Ek = Hij (wfc) - Hi;>
a;2 - o;^ + 2iC,r^r^k
Ck
(4.2)
where the constant Ck is given by the difference between the experimental value of the FRF and the modal contributions of all the other modes s ^ r. By introducing the following modal quantities as variables of the problem: Qjf — U)^ ,
O-p — ZC^fUJ'p ,
C^jr — H^irH^jr
(4.3)
Elements of Experimental Modal Analysis
9
Figure 7. Display of the FRF ^33 of a 3D0F system.
and a suitable weight function m =
9—-T—,
(4.4)
the error function can be linearized with respect to the assumed variables: Ek = —^
2 . 'o-/"^
^k= JiwkCijr + WkCk (ar " Cc;^ + ibrUJk) •
(4.5)
The procedure follows an iterative scheme: at the h-th iteration, the error is expressed as: Ek,h = iJkWk,h-\Cijr,h + 'Wk,h-lCk,h-l {Or^h " ^fe + ^W^h^k) (4.6) which is linear with respect to the three r-th modal unknowns since Wk and Ck are determined by the values at the step h — 1. The objective function is obtained by summing the square error in a frequency range in the neighborhood of ujr • On the basis
10
F. Vestroni and A. Pau
Figure 8. Free response and FRF of a simply supported beam.
of the least-squares method, the modal unknowns are determined. At each step, this procedure is extended to all meaningful resonances which appear in the response and, at the end of the process, all the modal parameters in the range of frequencies investigated are determined (Beolchini and Vestroni, 1994; Genovese and Vestroni, 1998; De Sortis et al., 2005). The second method presented is a frequency domain decomposition method and relies on the response to ambient excitation sources when the output only is available. The method is based on the singular value decomposition of the spectral matrix (Brincker et al., 2001). It exploits the relationship: Gyy (a;) = H (a;) Gxx (uj) H ^ (a.')
(4.7)
where Gxx{uj) (i?xi?, R number of inputs) and Gyy{u) {MxM, M number of measured responses) are respectively the input and output power spectral density matrices, and H(a;) is the frequency response function matrix (MxR). Supposing the inputs at the different points to be completely uncorrelated and white
Elements of Experimental Modal Analysis
11
noise, Gxx is a constant diagonal matrix G, independent of u. Thus: Gyy (a;) = G H {u) H ^ {u)
(4.8)
whose term jk can be written as, by omitting the constant G:
r=l \p=l
P
/
\g=l
9
/
In the neighborhood of the i-ih resonance, the previous equation can be written as: (4.10) R
where ^ (|)'^^ is a constant. By ignoring this term, the matrix Gyy can thus be expressed as the product of three matrices: G r o M = *Ai*^
(4.11)
which represents a singular value decomposition of the matrix Gyy, where: A,-
0 0
0... 0 0... 0 0... 0
(4.12)
The peaks of the first singular values indicate the natural frequencies of the system. In the neighborhood of the i-th peak of the first singular value, the first singular vector is coincident with the i-th eigenvector. This occurs at each j-th resonance, when the prevailing contribution is given by the j-th mode. This procedure, which had recently a great diffusion, was implemented in a commercial code (ARTeMIS).
5 Conclusions For SDOF and NDOF systems the knowledge of H{uj) provides a predictive model of the mechanical system in evaluating the response to any excitation. Moreover, it is possible to obtain experimental values of some components of H{uj) and then extract experimental values of modal parameters which are characteristic dynamic properties of the structure. Several methods in frequency and time domain are available to evaluate modal parameters from measured response to known and unknown excitation.
Bibliography ARTeMIS. modal software, www.svibs.com. G. C. Beolchini and F. Vestroni. Identification of dynamic characteristics of base-isolated and conventional buildings. In Proceedings of the 10th European Conference on Earthquake Engineering^ 1994.
12
F. Vestroni and A. Pau
S. Braun, D. Ewins, and S. S. Rao, editors. Encyclopedia of Vibration, 2001. Academic Press, San Diego. R. Brincker, L. Zhang, and P. Andersen. Modal identification of output-only systems using frequency domain decomposition. Smart Materials and Structures, 10:441-445, 2001. R. R. Craig. Structural Dynamics. John Wiley & Sons, 1981. A. De Sortis, E. Antonacci, and F. Vestroni. Dynamic identification of a masonry building using forced vibration tests. Engineering Structures, 27:155-165, 2005. D. J. Ewins. Modal Testing: theory, practice and application. Reaserch Studies Press, 2000. F. Genovese and F. Vestroni. Identification of dynamic characteristics of a masonry building. In Proceedings of the 1th European Conference on Earthquake Engineering, 1998. H. G. D. Goyder. Methods and applications of structural modelling from measured frequency response data. Journal of Sound and Vibration, 68:209-230, 1980. N.M.M. Maia and J.M.N. Silva. Theoretical and Experimental Modal Analysis. Reaserch Studies Press, 1997. L. Meirovitch. Principles and Techniques of Vibration. Prentice-Hall, 1997.
Damage Identification using Inverse Methods Michael I. Friswell Department of Aerospace Engineering, University of Bristol, Bristol BS8 ITR, UK.
[email protected] Abstract This chapter gives an overview of the use of inverse methods in damage detection and location, using measured vibration data. Inverse problems require the use of a model and the identification of uncertain parameters of this model. Damage is often local in nature and although the effect of the loss of stiffness may require only a small number of parameters, the lack of knowledge of the location means that a large number of candidate parameters must be included. This leads to potential ill-conditioning problems, and this topic is reviewed in this chapter. This chapter then goes on to discuss a number of problems that exist with the inverse approach to structural health monitoring, including modelling errors, environmental effects, damage localisation, regularisation, models of damage and sensor validation.
1 Introduction to Inverse Methods Inverse methods combine an initial model of the structure and measured data to improve the model or test an hypothesis. In practice the model is based on finite element analysis and the measurements are acceleration and force data, often in the form of a modal database, although frequency response function (FRF) data may also be used. The estimation techniques are often based on the methods of model updating, which have had some success in improving models and understanding the underlying dynamics, especially for joints (Friswell and Mottershead, 1995; Mottershead and Friswell, 1993). Model updating methods may be classified as sensitivity or direct methods. Sensitivity type methods rely on a parametric model of the structure and the minimisation of some penalty function based on the error between the measured data and the predictions from the model. These methods offer a wide range of parameters to update that have physical meaning and allow a degree of control over the optimisation process. The alternative is direct updating methods that change complete mass and/or stiffness matrices, although the updated models obtained are often difficult to interpret for health monitoring applications. These methods will be considered in more detail later. However it should be emphasised that a huge number of papers have been written on the application of inverse methods to damage identification, and this chapter aims to give an overview of the approaches rather than a complete literature review. This chapter will also consider some of the difficulties that occur when inverse methods are used for damage identification (Friswell, 2007; Doebling et al., 1998). The four stages of damage estimation, first given by Rytter (1993), are now well established as detection, location, quantification and prognosis. Detection is readily
14
M.I. Friswell
performed by pattern recognition methods or novelty detection (Worden, 1997; Worden et al., 2000). The key issue for inverse methods is location, which is equivalent to error locaUsation in model updating. Once the damage is located, it may be parameterised with a limited set of parameters and quantification, in terms of the local change in stiffness, is readily estimated. Prognosis requires that the underlying damage mechanism is determined, which may be possible using inverse methods using hypothesis testing among several candidate mechanisms. This questions is considered in more detail later in the chapter. However, once the damage mechanism is determined, the associated model is available for prognosis, and this is a great advantage of model based inverse methods. 1.1
Objective Functions
Friswell and Mottershead (1995) discussed sensitivity based methods in detail. The approach minimises the difference between modal quantities (usually natural frequencies and less often mode shapes) of the measured data and model predictions. This problem may be expressed as the minimization of J, where J(e) = | K - z ( ^ ) f = e^e
(1.1)
and e = Zm-z{e).
(1.2)
Here z ^ and z{6) are the measured and computed modal vectors, ^ is a vector of all unknown parameters, and e is the modal residual vector. The modal vectors may consist of both natural frequencies and mode shapes, although often mode shapes are only used to pair individual modes. If mode shapes are included then they must be carefully normalised, the sensor locations must be carefuly matched to the finite element degrees of freedom and weighting should be applied to Equation (1.1). Frequency response functions may also be used, although a model of damping is required, and the penalty function is often a very complicated function of the parameters with many local minima, making the optimisation very difficult, dos Santos et al. (2005) presents an example of such a method for damage in a composite structure. 1.2
Sensitivity Methods
Sensitivity based methods allow a wide choice of physically meaningful parameters and these advantages has led to their widespread use in model updating. The approach is very general and relies on minimising a penalty function, which usually consists of the error between the measured quantities and the corresponding predictions from the model. Parameters are then chosen that are assumed uncertain, and these are usually estimated by approximating the penalty function using a truncated Taylor series and iterating to obtain a converged solution. If there are sufficient measurements and a restricted set of parameters then the identification may be well-conditioned. Often some form of regularisation must be applied, and this is considered in detail later. Other optimisation methods may be used, such as quadratic programming, simulated annealing or genetic algorithms, but these are not considered further in this chapter. Problems will also arise
Damage Identification using Inverse Methods
15
if an incorrect or incomplete set of parameters is chosen, or even worse, if the structure of the model is wrong. The modal residual in Equation (1.1) is a non-linear function of the parameters and the minimization is solved using a truncated linear Taylor series and iteration. Thus the Taylor series is z ^ = Zj + Sj60j + higher order terms (1.3) where Zj = z{6j),
Sj = SiOj),
56j = em- Oj.
(1.4)
The matrix S^ consists of the first derivatives of the modal quantities with respect to the model parameters, index j denotes the j t h iteration and 6m is the parameter vector that gives the measured outputs. Standard methods exist to calculate the modal derivatives required (Priswell and Mottershead, 1995; Adhikari and Friswell, 2001). By neglecting higher order terms in Equation (1.3), an iterative scheme may be derived, using the hnear approximation, Szj = Sj56j (1.5) where 5zj = z ^ — Zj and 56j = 6j^i — 6j. Often, for damage location studies, only the residual and sensitivity matrix for the initial model are used. Avoiding iteration reduces the computation required, particularly where multiple parameter sets have to be estimated. However, particularly if the damage is severe, there is a risk that the wrong location is identified. As indicated above, one of the problems with sensitivity methods is the need for a parameteric model of the damage. Mottershead et al. (1999) proposed an approach where the system was constrained so that unknown stiffnesses are replaced with rigid connections. The constraint is not imposed physically but the behaviour inferred from the unconstrained measurements. The best fit between the measured and predicted data is obtained when the damage is located in the substructure that is made rigid. 1.3
Model Parameters
One of the key aspects of a model based identification method is the parameterisation of the candidate damage. Since inverse approaches rely on a model of the damage, the success of the estimation is dependent on the quality of the model used. The type of model used will depend on the type of structure and the damage mechanism, which leads to an increase either in local or distributed flexibility. The damage model may be simple or complex. For example, a cracked beam may be modelled as a reduction in stiffness in a large finite element or substructure, or alternatively using a very detailed model from fracture mechanics. Whether such a detailed model is justified will often depend on the requirements of the estimation procedure and the quality of the measured data. Using a measured modal model consisting of the lower natural frequencies and associated mode shapes will mean that only a coarse model of the damage may be identified. The simple example used for illustration will use element stiffnesses as the parameters and is the simplest form of equivalent model for the damage. More detailed models of damage will be considered in Section 3.
16 1.4
M.L Friswell Optimisation Procedures and Ill-Conditioning
When the parameters of a model are unknown, they must be estimated using measured data. Usually the measured response will be a non-linear function of the parameters. In these cases, minimizing the error between the measured and predicted response will produce a non-linear optimisation problem, with the usual questions about convergence and local minima. The most common approach is to linearise the residuals, obtain a least squares solution and iterate. If the identification problem is well posed then this simple approach will be adequate. The usual response to problems encountered in the optimisation is to try more advanced algorithms, but often the issue is that the estimation problem has not been posed correctly, and including some physical insight into the problem provides a much better solution. Probably the most important difficulty in parameter estimation is ill-conditioning. In the worst case this can mean that there is no unique solution to the estimation problem, and many sets of parameters are able to fit the data. Many optimization procedures result in the solution of linear equations for the unknown parameters. The use of the singular value decomposition (SVD) (Golub and van Loan, 1996) for these linear equations enables ill-conditioning to be identified and quantified. The options are then to increase the available data, which is often difficult and costly, or to provide extra conditions on the parameters. These can take the form of smoothness conditions (for example, the truncated SVD), minimum norm parameter values (Tikhonov regularization) or minimum changes from the initial estimates of the parameters (Hansen, 1992, 1994). Black box methods are often not considered as model based approaches. However any simulation of an input-output relationship must make some assumptions about the underlying process, and hence essentially has an underlying model. For example a neural network is essentially a very sophisticated curve fitting algorithm, and ill-conditioning is a major problem, evidenced by over-fitting and a lack of generalisation. The advantage of neural networks is that the class of input-output relationships that may be fitted is huge. However, better results will always be obtained if physical insight is used to guide the modelling and estimation process. Indeed there is often a need to reduce the number of input nodes to present to a neural network, and understanding is vital to obtain the correct feature extraction and data reduction. Another use of physical models is the generation of training or test data for these identification schemes. Typically experimental data for a sufficient range of events is diflacult or expensive to obtain. Since running a model many times is relatively easy and cheap, these simulations may be used to increase the quantity of the test data. However it is vital that this simulated data correctly reproduces the important features of the real structure, and hence requires a validated and, if necessary, updated model. Neural networks and genetic algorithms have been viewed as potential saviours for the solution of the difficult problems in damage location. Although these methods may be useful in some circumstances they do not deal with the root cause of the problem. Genetic algorithms have some advantage in finding a global minimum in very difficult optimisation problems, particularly where there are many local minima as is often the case in damage location. That said, the method still requires that the dynamics of the structure changes sufficiently and predictably enough for the optimisation to be
Damage Identification using Inverse Methods
17
meaningful. The crucial decision and difficulty is what to optimise, not the optimisation method used. Neural networks are able to treat damage mechanisms implicitly, so that it is not necessary to model the structure in so much detail. The method can also deal with non-linear damage mechanisms easily. Models are still required to provide the training cases for the networks, and this is their major problem. There will always be systematic errors between the model used for training and the actual structure. For success, neural networks require that the essential features in the damaged structure were represented in the training data. The robustness of networks to these errors has not been tested sufficiently. Of course, the other major problem with both genetic algorithms and neural networks is that they require a huge amount of computation for structures of practical complexity, although these methods are well suited to parallel computation. 1.5
Problems and Errors in Damage Identification
The discussion thus far has indicated some of the problems with damage identification. There are always errors in the measured data and the numerical model that affect all of the algorithms. These errors, and the adequacy of the data, are now discussed. Damage identification algorithms should always be tested on realistic experimental examples, as many methods that work well on simulated data often fail due to the problems highlighted in this section. As a first step, methods may be tested using simulated data, but even then realistic systematic errors should be incorporated. Modelling errors One of the major problems in damage location is the reliance on the finite element model. This model is also an important strength because the very incomplete set of measured data requires extra information from the model to be able to identify damage location. There will undoubtedly be errors even in the model of the undamaged structure. Thus if the measurements on the damaged structure are used to identify damage locations, the methods will have great difficulty in distinguishing between the actual damage sites and the location of errors in the original model. If suitable parameters are not included to allow for the undamaged model errors then the result will be a systematic error between the model and the data. Identification schemes generally have considerable difficulty with systematic errors. It is very likely that the original errors in the model will produce frequency changes that are far greater than those produced by the damage. There are two basic approaches to reducing this problem, although both rely on having measured data from an undamaged structure. The first is to update the finite element model of the undamaged structure to produce a reliable model (Priswell and Mottershead, 1995). Obviously the quality of the damage location assessment is critically dependent upon the updated model being physically meaningful (Friswell et al., 2001; Link and Friswell, 2003). Generally, this requires model validation using a control set of data not used for the updating. The second alternative uses differences between the damaged and undamaged response data in the damage location algorithm (Parloo et al., 2003; Titurus et a l , 2003b). To first order, any error in the undamaged model of the structure that is also present in the damaged structure will be removed. This does rely on the structure remaining unchanged, except for the damage.
18
M.L Friswell
between the two sets of measurements. Another potential source of error is the mismach between the measurement locations and the model degrees of freedom. Such a mismatch makes the direct comparison of frequency response functions and mode shapes impossible, and the generation of residuals, inaccurate. The magnitude of the errors involved will depend on the mesh density in the sensor region and the complexity of the mode shapes. The best solution is to ensure nodes in the model exist at the sensor locations. Alternatively interpolation techniques may be used. Environmental and other non-stationary effects One very difficult aspect of damage assessment is the change in the measured data due to environmental effects. This is one undesirable non-stationary effect and makes damage location very difficult. Of course progressive damage is also a non-stationary phenomenon, and damage can be difficult to identify if other non-stationary effects are also present. Typical environmental effects are demonstrated by highway bridges, especially those constructed using concrete, which have been the subject of many studies in damage location. For example, temperature changes can cause the stiffness properties of a bridge to change significantly, and the difficulty is to predict the effects of temperature from readily available measurements. Peeters and de Roeck (2001) reported on measurements of the Z24 bridge over a whole year and suggested a black box model to predict the temperature variation. Sohn et al. (1999) considered the effect of temperature on the Alamosa Canyon Bridge. Sohn et al. (2002) used a combination of time series analysis, neural networks and statistical inference to determine damage state for structures affected by the environmental conditions. Mickens et al. (2003) corrected frequency response function measurements by assuming the temperature affected the global stiffness of the structure. On a highway bridge, the changing traffic conditions cause different mass loading effects that can change the natural frequencies by as much as 1% (Zhang et al., 2002). There are further difficulties with highway bridges because they are highly damped with low natural frequencies. They are in a noisy environment and are difficult to excite. The frequency resolution in the measurements is invariably quite low, leading to considerable difficulties in detecting small frequency changes due to damage. Typical of environmental effects are those in highway bridges. These bridges have been the subject of many studies in damage location, but in the UK, where most bridges are constructed using concrete, such identification has considerable problems with changes due to environmental factors (Wood, 1992). For example, concrete absorbs considerable moisture during damp weather, which considerably increases the mass of the bridge. Temperature changes the stiffness properties of the road surface, known as the black-top^ significantly. On a hot summer's day in the UK, the road surface will provide little stiffness, but on a cold winter's day the stiffness contribution is considerable. The difficulty is trying to predict the effects of temperature and moisture absorption from readily available measurements. Figure 1 shows the variation of the first 4 natural frequencies of a concrete highway bridge in Birmingham, UK with soffit temperature (Wood, 1992). Soffit temperature is the variable that correlates best with the frequencies, but even then relatively large, unexplained variations in frequency occur.
19
Damage Identification using Inverse Methods 20 18 16 14 12 10 5
10
15
o
20
Soffit Temperature (C) Figure 1. Variation of the 4 lowest natural frequencies of the New Haymills Bridge, Birmingham, UK.
The effect of frequency range The range of frequencies employed in damage location has a great influence on the resolution of the results and also the physical range of application. The great advantage in using low frequency vibration measurements is that the low frequency modes are generally global and so the vibration sensors may be mounted remotely from the damage site. Equally fewer sensors may be used. The problem with low frequency modes is that the spatial wavelengths of the modes are large, and typically are far larger than the extent of the damage. The spatial resolution of the damage identification scheme requires that there is a significant change in response between two adjacent potential damage sites. If low frequency modes are used then this resolution is closely related to the spatial wavelengths of the modes. Using high frequency excitation uses very local modes which are able to accurately locate damage, but only very close to the sensor and actuator position. Estimating accurate models at these high frequency ranges is also very difficult, and often changes in the response are used for damage identification. For example. Park et al. (2000, 2001) used changes in measured impedance to identify damage in civil structures and pipeline systems. Schulz et al. (1999) used high frequency transmissability to detect delaminations in composite structures. Moving to even higher frequencies can also yield good results. Acoustic emission (Rogers, 2001) is a transient elastic wave, typically in the region of 50 to 500kHz, and is able, for example, to detect the energy released when cracks propagate. One approach to damage assessment is to use physical models to deduce quantitative relationships between measured acoustic emission signals and the damage mechanism that cause them. Significant research has been undertaken to obtain a physical understanding of various source mechanisms (Scruby and Buttle, 1991) and the radiation pattern of bulk shear and longitudinal acoustic waves that they produce (Ono, 1991). The diflSculty in using these models in inverse estimation procedures is the accuracy of these high frequency
20
M.I. Friswell
models, and the huge computational requirements. In recent years a pattern recognition philosophy has dominated, that relies on using large databases of empirical data from which correlations between measured acoustic emission signals and damage mechanisms are inferred. Many advanced signal processing algorithms have been employed to interpret experimental data. Damage location is often determined using time of flight methods, that require the events to be well separated in time, the wave speed to be approximately equal in all parts of the structure, and the effects of reflection and refraction to be insignificant. Examples of the use of acoustic emission for health monitoring are given by Rogers (2001), Atherton et al. (2004) and (Holford et al., 2001). Damage magnitude A frequent problem that arises in model-based vibration-based damage detection, whether parametric or non-parametric, is the need for a very accurate mathematical model, so that it correctly captures the actual structural dynamic behaviour in some predetermined frequency range. Often in structural health monitoring the changes in the measured quantities caused by structural damage are smaller than those observed between the healthy (i.e. undamaged) structure and the mathematical model. Consequently, it becomes almost impossible to discern between inadequate modelling and actual changes due to damage. There are two alternative approaches to this problem. The first is to update the healthy model so that the correlation between the model and the measured data is improved. This approach requires that the errors that remain after updating are smaller in magnitude than the changes due to the damage. Furthermore the changes to the model should be physically meaningful, so that the updating process corrects actual model errors, and doesn't merely reproduce the measured data. The second approach is based on the use of (relative) differences between data measured on healthy and potentially damaged structure. In this case, assuming that the only changes in the structure are due to damage, the problem may be reduced to finding those parameters that reproduce the measured changes. Non-linearity Many forms of damage cause a change in the stiffness non-linearity that qualitatively and quantitatively affects the dynamic response of a structure. For example, Nichols et al. (2003a,b) used the features of the chaotic response of a structure to detect changes in a joint. Adams and Nataraju (2002) gave a variety of features based on the non-linear dynamic response. Kerschen et al. (2003) considered model based estimation methods and identified the form of nonlinearity that is most hkely present in the measured data. Meyer and Link (2003) identified a parametric non-linear model using harmonic balance and a model updating approach. A breathing crack, which opens and closes, can produce interesting and comphcated non-linear dynamics. Brandon (1998) and Kisa and Brandon (2000) gave an overview of some of the techniques that may be applied. Many techniques to analyse the resulting non-linear dynamics are based on approximating the bilinear stiffness when the crack opens and closes. Linear approaches to damage estimation approximates a local reduction in the stiffness matrix of the beam. Since the non-linearity introduced by a crack is often weak, many of the common testing techniques will tend to linearise the response (Friswell and Penny, 2002). Sinusoidal forcing will tend to emphasise the non-hnearity, and damage detection
Damage Identification using Inverse Methods
21
methods based on detecting harmonics of the forcing frequency have been proposed (Shen, 1998). In rotor dynamic appHcations these approaches are useful because the forcing is inherently sinusoidal (Dimarogonas, 1996). However in structural health monitoring applications this approach requires considerable hardware and software to implement, and also requires a lengthy experiment. Johnson et al. (2004) used a transmissibility approach that was insensitive to boundary condition non-linearities. Neild et al. (2003) investigated the potential of a time frequency analysis procedure to identify damage in concrete beams. Although using the non-linear response has a huge potential in health monitoring, model based inverse approaches have a number of difficulties because of the high number of degrees of freedom required, and therefore the computational burden imposed. In practice, any realistic multi degree of freedom non-linear analysis would have to be based on a reduced order model of the structure. Furthermore, many of the difficulties outlined in this section for linear systems, are also a problem for non-linear systems. Strength vs. stiffness The philosophy of damage detection using measured vibration data is based on the premise that the damage will change the stiffness of the structure. In some instances there is a significant difference between strength and stiffness. Indeed, estimating the remaining useful life of a component based on conclusions from a dynamic analysis is very difficult. For example, a concrete highway bridge will have steel reinforcement cables running in channels in the concrete. The cables are tensioned, either before or after the concrete has set, to ensure that the concrete remains in compression. One major failure mechanism is by the corrosion of these cables. Once the cables have failed the concrete has no strength in tension and so the bridge is liable to collapse. Unfortunately the stiffness of the bridge is mainly due to the concrete, and so the progressive corrosion of the cables is very difficult to identify from stiffness changes. Essentially the dynamics of the bridge do not change until it collapses. 1.6
The Role of Simulation and Physical Testing
Many of the algorithms suggested for damage location are tested on simulated data. It is necessary to fully test any method on both simulated and real data. The simulated tests are able to fully exercise the location methods, with the benefit that the answer is known. In simulation, far more damage cases may be used and the effect of errors may be fully investigated. The need for real testing arises because experimental work always produces errors and problems that are unexpected. For simulation to be useful, the errors that might be expected in real structures must be simulated. Thus, adding random noise to a model of the structure and then using the same model to identify the damage in not enough! Most identification schemes are able to cope very well with random noise, and although such simulations are important parts of the overall performance assessment of an algorithm, they are not sufficient. It is vital that systematic type errors are included in the simulation. Thus, discretisation errors may be included by generating the simulated measurements using a fine finite element model; the damage mechanism introduced to generate the measurements may be different to those modelled for the identification; or boundary conditions on the structure could be changed between the measured data set
22
M.I. Friswell
and the identification. 1.7
A Simple Cantilever Beam Example
A simulated cantilever beam example will be used to demonstrate some of the problems. Although the example is somewhat artificial it will highlight how easily methods fail even on very simple structures. Any practical method would have to be robust and should therefore succeed on simple structures, even though some systematic errors are included. This example also demonstrates the use of simulation in damage identification. Not all of the methods are tried and this example is not supposed to represent an extensive scientific evaluation of the methods. Its purpose is for illustration. The beam has a cross section of 25mm x 50mm, a length of Im and is assumed to be rigidly clamped at one end. Only motion in the plane of the thinner beam dimension is considered. The beam has a Young's modulus of 210GN/m and a mass density of 7800kg/m^ The first test of any method is its application to a simulated example with no noise or systematic errors. Any parameter changes in the model should be identified exactly. The simulated measurements are assumed to be the relative changes in the lower natural frequencies of the beam and are taken from a model with 20 elements. The undamaged natural frequencies are taken from the uniform beam, whilst the damaged frequencies are derived from a model where the stiffness of element 4 has been reduced by 30%. Table 1 gives the damaged and undamaged natural frequencies, showing that the 30% damage only results in a 2.4% change in natural frequency at most. These small frequency changes are typical in damage location and are one of the major diSiculties in the identification of the location of damage. Measurement noise, environmental factors and structure nonstationarity can easily lead to incorrect conclusions on damage location. Table 1. Natural frequencies of the simulated undamaged and damaged beam. Mode No. 1 2 3 4 5 6 7 8
Undamaged (Hz) 20.96 131.3 367.7 720.6 1191 1780 2487 3313
Damaged (Hz) 20.45 131.1 366.6 711.3 1172 1762 2479 3303
Difference(%) 2.39 0.15 0.31 1.29 1.61 1.02 0.32 0.30
The standard sensitivity approach based on modal data will now be used to identify the damage. The set of candidate parameters is chosen to be relatively large and consists of the stiflFness of each of the 20 elements. If the relative changes to the first 8 natural frequencies are used as the measurements then the identification of the parameters is under determined. In this case some form of regularisation must be employed. Figure 2 shows the change in element stiffness required to reproduce the damaged natural frequencies, using a minimum norm constraint on the parameter changes. Although the largest
23
Damage Identification using Inverse Methods
stiffness change occurs at element 4 the identified damage is spread over the whole beam, and there are some significant increases in stiffness. Note that this is the ideal case with no measurement noise or modelling errors. Suppose that, by some means, the damage is known to be somewhere in the eight elements closest to the fixed end. The number of parameters is now reduced to eight, the same as the number of natural frequencies. There is now a unique solution to the estimation problem and this solution is given in Figure 3. Note that the stiffness of elements 9 to 20 cannot change, but are included in Figure 3 for easy comparison. The damage has clearly been correctly located to element 4. However the magnitude of the damage is incorrect because the estimation is based on the sensitivity matrix which is a linear approximation to the residual. The other seven parameters are non-zero for the same reason. 10
0
-10
-20
1
5
10 Element Number
15
20
Figure 2. The change in element stiffness estimated for the cantilever beam example with no noise and a minimum norm constraint.
2 Regularisation The advantages of sensitivity type model updating methods have been highlighted in this chapter. However there are significant differences in the application of these methods in model updating and damage location, which necessitates different methods of regularisation. In both cases the number of potential parameters is very large and the estimation process is likely to be ill-conditioned unless the physical understanding can be used to introduce extra information. In model updating, the number of parameters may be reduced by only including those parameters that are likely to be in error. Thus if a frame structure is updated, the beams are hkely to be modelled accurately but the joints are more difficult to model. It would therefore be sensible to concentrate the uncertain parameters to those associated with the joints. Even so, a large number of potential parameters may be generated, the measurements may still be reproduced and the parameters are unlikely to be identified
24
M.I. Friswell
0
-20
•-40
1
5
10 Element Number
15
20
Figure 3. The change in element stiffness estimated for the cantilever beam example with no noise, but only 8 non-zero parameters.
uniquely. In this situation all the parameters are changed, and regularisation must be applied to generate a unique solution (Friswell et al., 2001). Regularisation generally applies extra constraints to the parameter estimation problem to ensure a unique solution. Applying the standard Moore-Penrose pseudo inverse is a type of regularisation where the parameter vector with the minimum norm is chosen. The parameter changes may be weighted separately to give a weighted least squares problem, where the penalty function is a weighted sum of squares of the measurement errors and the parameter changes. Such weighting may also be extended to include minimising the difference between equivalent parameters that are nominally equal in different substructures such as joints. Although using parametric models can reduce the number of parameters considerably, for damage location there will still be a large number of parameters. Most regularisation techniques rely on minimum norm type solutions that will tend to spread the identified damage over a large number of parameters. Using subset selection, where only the optimum subset of the parameters are used for the estimation (Friswell et al., 1997), has been used for model updating and also for damage location. 2.1
Tikhonov Regularisation
The treatment of ill-conditioned, noisy systems of equations is a problem central to finite element model updating (Ahmadian et al., 1998). Such equations often arise in the correction of finite element models by using vibration measurements. The regularisation problem centres around the linear equation. Ax = b
(2.1)
where x is a vector of the m parameter changes we wish to determine, and b is a vector of n residual quantities derived from the measured data and the current estimate of the model. Note that the iteration index j has been dropped from the expressions
Damage Identification using Inverse Methods
25
in Section 1.2, and 59 has been replaced by x. In model updating the relationship between the measured output (for example, natural frequencies, mode shapes, or the frequency response function) is generally non-linear. In this case the problem is linearised using a Taylor series expansion and iteration performed until convergence. When b is contaminated with additive, independent random noise with zero mean, it is well known that the least-squares solution, X£,5, is unique and unbiased provided that rank (A) = m. When A is close to being rank deficient then small levels of noise may lead to a large deviation in the estimated parameters from its exact value. The solution is said to be unstable and Equation (2.1) is ill-conditioned A different problem occurs when m > n so that Equation (2.1) is under-determined and there are an infinite number of solutions. The Moore-Penrose pseudo-inverse in the form, XL5-A^[AA^]"'b (2.2) provides the solution of minimum norm, as does singular value decomposition (SVD). For the case when rank (A) = r < min (m, n), the SVD will again result in the minimum norm solution. This is a form of regularisation which has been widely apphed in the model updating community. Unfortunately minimum norm solutions rarely lead to physically meaningful updated parameters. Side Constraints Model updating often leads to an ill-conditioned parameter estimation problem, and an effective form of regularisation is to place constraints on the parameters. This could be that the deviation between the parameters of the updated and the initial model are minimised, or differences between parameters could be minimised. For example, in a frame structure a number of 'T' joints may exist that are nominally identical. Due to manufacturing tolerances the parameters of these joints will be slightly different, although these differences should be small. Therefore a side constraint is placed on the parameters, so that both the residual and the differences between nominally identical parameters are minimised. Thus if Equation (2.1) generates the residual, the parameter is sought which minimises the quadratic cost function,
J (x) = IIAx - b f + A^ IJCx - d f
(2.3)
for some matrix C, vector d and regularisation parameter A. The regularisation parameter is chosen to give a suitable balance between the residual and the side constraint. For example, if there were only two parameters, which were nominally equal, then C = [l-1]
(2.4)
Minimising Equation (2.3) is equivalent to minimising the residual of A AC
M^U
(2.5)
Equation (2.5) then replaces Equation (2.1), although with the significant difference that Equation (2.5) is generally over-determined, whereas Equation (2.1) if often underdetermined. The constraints should be chosen to satisfy Morozov's complementation
26
M.I. Friswell
condition rank | ^
= m
(2.6)
which ensures that the coefficient matrix in Equation (2.5) is full rank. The Singular Value Decomposition may be written in the form,
The singular value decomposition (SVD) of A m
A = USV^ = Y, ^i^i^I where U = [uiU2 ...Un] and V = [viV2 . . . v^] are nxn and mxm and S = diag (cTi, cr2,..., am)
(2-7) orthogonal matrices (2.8)
where the singular values, cr^, are arranged in descending order ((TI > 0-2 ^ • • • > ^m)- In ill-posed problems two commonly occurring characteristics of the singular values have been observed; the singular values decay steadily to zero with no particular gap in the spectrum, and the left and right singular vectors u^ and v^ tend to have more sign changes in their elements as the index i increases. The solution for the parameters using the SVD is
Thus the components of A corresponding to the low singular values have only a small contribution to A but a large contribution to the estimated parameters. The elements of these singular vectors (corresponding to the low singular values) are also generally highly oscillatory. Equation (2.9) shows the noise will be amplified when cr^ < u ^ b , and this may be used to decide where to truncate the singular values. If A does not contain noise then the singular values will decay to zero whereas the u ^ b terms will decay to the noise level. Ahmadian et al. (1998); Hemez and Farhat (1995) consider this approach in more detail. The standard SVD is incapable of taking account of the side constraint, as this requires the generalised SVD. Space does not permit a full explanation of the generalised SVD, and the reader is referred to Hansen (1994) for more complete detail of the decomposition. In Equation (2.5), A and C are decomposed as A = U
I 0
0 S
"-1
r« _= \7A4-1 C V [r0n M ] XY- -i i
(2.10)
where X is a non-singular mxm matrix. U and V are n x m and p x p respectively, and their columns are orthogonal (but they are not related to the matrices U and V of the standard SVD) and n>m>p. The matrices E and M are S = diag (^1,^2, ....oTp)
M = diag (//i, /X2, • • •, A^p)
(2.11)
Damage Identification using Inverse Methods
27
where 1 > cri > (J2 > •.. > cr^ > 0 and 0 < /xi < /X2 < . •. < /Xp < 1, and ai and /Xi are normahsed so that,
(2.26)
where
o^i = {^l^i)/{^W)-
(2-27)
A second parameter may now be selected by means of the modified problem defined by Equation (2.26), where j ^ ji. Further parameters may be selected in the same way. An algorithm is thus created to search for the best parameter subset, denoted by [xj,,Xj2,. • •,%p], that minimises the cost function, jp —
"-Ei=l
(2.28)
Damage Identification using Inverse Methods
31
This cost function is also employed in Efroymson's algorithm for forward subset selection, which focuses on adding or removing parameter selections from chosen subsets. Thus the number of candidate damage locations may be controlled based on the expected reduction in the residual (Millar, 1990; Priswell et al., 1997). Since only single damage location cases will be examined in detail here this subject will not be considered further. Weighting The final theoretical aspect is the need for weighting when Equation (1.5) is used for damage location, and two types of weighting will be considered. First, weighting is needed to handle the different numerical values corresponding to the different modal quantities. Thus, only relative, or percentage changes in the modal quantities, due to damage, will be employed. The second type of weighting arises as a result of combining two different entities in the sensitivity matrix, namely natural frequencies and mode shapes for complete damage location. The experimental origin of the measurements means that the errors in mode shape estimation are usually greater than the errors in natural frequency estimation. This weighting is employed in the calculation of the subspace angles between the vector 5z and the columns of the matrix A, Equation (2.23), and is based on the weighted scalar product (Knyazev and Argentati, 2002). The procedure is also called the scalar A-based product and has its origins in statistics (note that the A in the name of this product has nothing to do with the matrix A in Equation (2.21)). Knyazev and Argentati (2002) studied this scalar product in the context of the numerically stable computation of principal angles between two linear subspaces. The scalar A-based inner product is defined as (x, y ) ^ = (x, Awy)
= y^ AH/X
(2.29)
where x , y G 3?'^ are vectors, and Aw G SR^^'^ is a symmetric, positive definite matrix. A-based vector and matrix norms, ||.. .||^, may be defined as V(x,x),
A^/^x
l|B|U =
(2.30)
where B G SR"^^"^ is an arbitrary matrix. The applicability of this type of product for damage location based on the additional use of mode shape sensitivities in S and mode shape differences due to damage will be studied for an experimental, geometrically symmetric structure in a later section. Since 5z is derived from experimental data, and assuming that the mass distribution does not change with damage, no additional scaling of individual mode shapes, with respect to other modes, will be employed. Since the natural frequencies are measured much more accurately than the mode shapes, the natural frequencies should be used to determine the candidate damage locations. The A weighting on the mode shapes is then used for geometrically symmetric structures to ensure that the most likely damage location from among the candidate locations identified from the natural frequencies is chosen. The weighting of the mode shapes is increased until a perceptible difference occurs between these candidate locations, but is kept as low as possible to reduce the effect of the noise on the mode shapes.
32
M.I. Friswell
2.3
The Simple Cantilever Beam Example Revisited
The simple cantilever beam example of Section 1.7 will be used to demonstrate some of the properties of the methods given in this section. Candidate parameters now include element mass, and discrete mass and springs, as well as the element stiffness. Figure 4 show^s the angles between the columns of the sensitivity matrix of the initial finite element model and the vector of the relative changes in the first 8 natural frequencies due to the damage. Clearly the column relating to the stiffness of element 4 has a small angle, although it is not zero because the method is based on a first order approximation and the extent of the damage (30%) is large. Changing the mass of element 17 is also able to model the measured changes accurately. This is a problem that relates to the symmetry of the beam, and the fact than no spatial information is incorporated into the measurements. Mode shapes could also be incorporated into the measurement vector, although the accuracy with which they could be measured may be insufficient to show a change in mode shape due to damage. This is an example of the more general problem, where damage or changes of parameters at more than one location causes the same changes in the lower natural frequencies.
m
Element Stiffiiess
Discrete Stiffness
Element Mass
Discrete Mass
60 40 20 n
£•11
2
3
4
5
6
7
8 9 10 11 12 13 14 15 16 17 18 19 20 Element or Node Number
Figure 4. Subspace angles for a 30% change to the stiffness of element 4. Subset selection is next demonstrated on an example where damage is introduced at 2 locations. A 0.1kg mass is added to node 12, in addition to the 30% stiffness change to element 4. This example does not include any measurement noise or modelhng errors. Table 2 shows the results when the best subsets of 1, 2 and 3 parameters are chosen. The parameters are specified by type and element or node number. Thus {pA)-^^ is the mass / unit length of element 17, (EI)^ is the stiffness of element 4, ki is a discrete spring at node 1 and mi2 is a discrete mass at node 12. At each stage the 2 best parameters are chosen. The residuals under the first 2 parameters relate to the values when a subset of size 1 or 2 is selected. Also shown are the residuals after convergence based on optimising the values of the chosen parameters. From the values of the residuals, it is clear that the two correct parameters should be selected. A number of other parameter subsets have small residuals and the addition of random noise would make the selection of the best subset more difficult.
33
Damage Identification using Inverse Methods Table 2. The selection of three parameters for the beam example. Parameter 1 Res- Convidual erged (/>^)l7
154.6
(EI),
154.7
Parameter 2 Res- Convidual erged 0.782 1.49 mi2
160.4 mg
8.70
4.76
mi2
1.49
0.000
ms
8.70
5.40
160.5
Parameter Residual 1.21 mg 1.48 (P^)l2 1.21 mi2 M ) l 2 8.68 1.20 mg 1.48 (P^)l2 1.20 mi2 8.69 (M)l2
3 Converged 0.701 0.286 0.701 4.75 0.000 0.000 0.000 5.35
3 Parameterisation of Models of Damage Damage usually causes a reduction in the local stiffness of the structures. One option is to model this a reduction in stiffness at the element or substructure level. This equivalent modelling approach is often sufficient for the identification of local damage using low frequency vibration measurements. This section considers more detailed models of damage that have parameters that may be identified using inverse methods. 3.1
Crack Models
The modelling of cracks in beam structures and rotating shafts has been a significant research topic. The models fall into three main categories; local stiffness reduction, discrete spring models, and complex models in two or three dimensions. Dimarogonas (1996); Ostachowicz and Krawczuk (2001) gave comprehensive surveys of crack modelling approaches. The simplest methods for finite element models reduce the stiffness locally, for example by reducing a complete element stiffness to simulate a small crack in that element (Mayes and Davies, 1984). This approach suffers from problems in matching damage severity to crack depth, and is affected by the mesh density. An improved method introduces local fiexibility based on physically based stiffness reductions, where the crack position may be used as a parameter for identification purposes. The second class of methods divides a beam type structure into two parts that are pinned at the crack location and the crack is simulated by the addition of a rotational spring. These approaches are a gross simplification of the crack dynamics and do not involve the crack size and location directly. The alternative, using beam theory, is to model the dynamics close to the crack more accurately, for example producing a closed form solution giving the natural frequencies and mode shapes of cracked beam directly or using differential equations with compatible boundary conditions satisfying the crack conditions (Christides and Barr, 1984; Sinha et al., 2002; Lee and Chung, 2001). Friswell and Penny (2002) compared several of the simple cracks models that may be used for health monitoring, for both the linear and non-linear response. Alternatively two or three dimensional finite element meshes for beam type structures with a crack may be used. Meshless approaches
34
M.L Friswell
may also be used, but are more suited to crack propagation studies. No element connectivity is required and so the task of remeshing as the crack grows is avoided, and a growing crack is modelled by extending the free surfaces corresponding to the crack (Belytschko et al., 1995). However the compuational cost of these meshless methods generally exceeds that of conventional finite element analysis (FEA). Rao and Rahman (2001) avoided this difficulty by coupling a meshless region near the crack with an FEA model in the remainder of the structure. The two and three dimensional approaches produce detailed and accurate models but are a complicated and computational intensive approach to model simple structures like beams, and are unlikely to lead to practical algorithms for damage identification. Models of open cracks Although the geometry of a crack can be very comphcated, the contention in this paper is that for low frequency vibration only an effective reduction in stiffness is required. Thus, for comparison, a simple model of an open crack, which is essentially a saw cut, will be used. This will allow the comparison of models using beam elements, with those using plate elements. Only a selection of beam models will be used, that illustrate the fact that many beam models are able to model the effect of the crack at low frequencies. Two standard approaches using beam elements are shown in Figure 5. In the first approach, the stiffness of a single element is reduced, which requires a fine mesh, and also the derivation of the effect of a crack on the element stiffness. In the second approach, the beam is separated into two halves at the crack location. The beam sections are then pinned together and a rotational spring used to model the increased flexibility due to the crack. Translational springs may also be used in place of the pinned constraint. The major difficulties with this approach is that a finite element node must be place at the crack location, requiring remeshing for health monitoring applications, and the relationship between the spring stiffness and crack depth needs to be derived.
Reduction in Element Stiffness
Pinned Joint at Crack Location Figure 5. Simple crack models for beam elements.
Damage Identification using Inverse Methods
35
For illustration, the open crack will be modeled using plate elements. The geometry is modeled by removing elements where the crack is located. Figure 6 shows this in the case of plate elements, and shows the side view of the mesh used. Clearly more complex methods may be used, and the review papers quoted earlier give further details.
h Figure 6. A simple crack model using plate elements.
The approach of Christides and Barr Clearly some of the material adjacent to the crack will not be stressed and thus will offer only a limited contribution to the stiffness. The actual form of this increased flexibility is quite complicated, but in this paper we approximate this phenomenon as a variation in the local flexibility. In reality, for a crack on one side of a beam, the neutral axis will change in the vicinity of the crack, but this will not be considered here. Shen and Pierre (1994); Carneiro and Inman (2001, 2002) have extended this approach to consider single edge cracks. Christides and Barr (1984) considered the effect of a crack in a continuous beam and calculated the stiffness, £ / , for a rectangular beam to involve an exponential function given by
EI{x)
Eh 1 -f- Cexp (—2a |a: — o^d 1^
(3.1)
w{d — dc) W^d^ are the second moment of where C = {IQ ~ Ic) /h- h = -j^r and Ic = areas of the undamaged beam and at the crack, w and d are the width and depth of the undamaged beam, and dc is the crack depth, x is the position along the beam, and Xc the position of the crack, a is a constant that Christides and Barr estimated from experiments to be 0.667. The inclusion of the stiffness reduction of Christides and Barr (1984) into a finite element model of a structure, using beam elements, is complicated because the flexibility is not local to one or two elements, and thus the integration required to produce the stiffness matrix for the beam would have to be performed numerically every time the crack position changed. Furthermore, for complex structures, without uniform long beams, Equation (3.1) would only be approximate. Sinha et al. (2002) used a simplified approach, where the stiffness reduction of Christides and Barr was approximated by a triangular reduction in stiffness. An example of this approximation is shown in Figure 7, for a crack of depth 5%, located at x = 0. The advantage of this simplified model is that the stiffness reduction is now local, and the stiffness matrix may be written as an explicit function of the crack location and depth. For cracks of small depth a good approximation to the length of the beam influenced by the crack is 2d/a.
M.I. Friswell
36
1
0.95
0.9
0.85 -
4
-
2
0
2
4
xld Figure 7. The variation in beam stiffness for the approaches of Christides and Barr (1984) (sohd Hne) and Sinha et al. (2002) (dashed hne).
Fracture mechanics approach An alternative approach is to estimate the increased flexibihty caused by the crack, using empirical expressions of stress intensity factors from fracture mechanics. Lee and Chung (2001) gave such an approach based on the relationships given by Tada et al. (1973). Only a summary of the relevant equations will be given here. The element stiffness matrix is given by (3.2)
K, = : T ' C - ^ T where the transformation, T, is -1 0
0 1 -1 0
0 1
(3.3)
The flexibility matrix, C, for an element containing the crack in the middle, is given by 6EI
2il Ml 3£? %L
+
ISTT (1 -
Ewd?
v'^)
2L
24 4
]
rdc/d
/3Ffil3)d(3.
(3.4)
where £e is the element length and u is Poisson's ratio. Fj {(3) is the correction factor for the stress intensity factor, and may be approximated as FiiH)
/tan (7r/3/2) 0.923 + 0.199 [1 - sin (7r/3/2)]^ 7r/?/2 cos (7r/3/2)
(3.5)
This formulation does give the stiffness matrix of the element containing the crack explicitly in terms of the crack depth. There are two difficulties with using this approach for structural health monitoring. The main problem is that the crack is located at the centre of the element, requiring that the finite element mesh be redefined as the crack moves. Furthermore the stiffness matrix of the crack is a complicated function of the crack depth, and does not depend on the crack location explicitly.
37
Damage Identification using Inverse Methods
A numerical comparison of the models The approaches to crack modelhng will be compared using a simple example of a steel cantilever beam Im long, with cross section 25 X 50mm. Bending in the more flexible plane is considered. The crack is assumed to be located at a distance 200mm from the fixed end, and has a constant depth of 10mm across the beam width. The beam is modelled using 20 Euler-BernouUi beam elements, and gives the natural frequencies shown in Table 3. For the plate elements the length is split into 401 elements and the depth into 10 elements. Thus the elements are approximately 2.5mm square. A large number of elements is required because an element with linear shape functions is used. Table 3 shows the estimated natural frequencies using the Quad4 element in the Structural Dynamics Toolbox (Balmes, 2000). Table 3. Natural frequencies (in Hz) for the undamaged beam. Number DoF
Modes
1 2 3 4 5
Beam 40 20.709 129.78 363.40 712.16 1177.4
Plate 13233 20.707 129.39 360.62 701.96 1150.6
The damaged beam was also modelled using the approaches discussed earlier, and the results are shown in Table 4. The beam models all contain 20 elements, and the nodes are arranged such that the crack occurs in the middle of an element. Of course in the case of the discrete rotational spring a node is placed at the crack location. The reduction in the element stiffness is adjusted so that the percentage change in the first natural frequency is the same as that for the plate model. The other beam models are adjusted in a similar way. In the plate model, the crack is simulated by removing 4 elements and thus represents a saw cut 10 mm deep. The row of elements below the crack is also made thinner, so that the crack has negligible width. The differences in the lower natural frequencies are very similar for all models, and these differences are smaller than the changes that would occur due to small modelling errors, or changes due to environmental effects. Of course the accuracy at higher frequencies becomes less since the modes are influenced more by local stiffness variations. Comparison with experimental results The previous section has shown that the natural frequencies predicted from different models are very close. Of course the question is whether the differences in these predictions are smaller than the measurement errors. As a demonstration the example of Rizos et al. (1990) will be used. Kam and Lee (1992) and Lee and Chung (2001) also used these results. The example is a steel cantilever beam of cross-section 20 x 20mm and length 300mm. Table 5 shows the measured and predicted frequencies of the uncracked beam. Rizos et al. (1990) propagated cracks at a number of different positions and depths, but here only a crack 80mm from the cantilever root, and depths of 2 and 6mm will be considered. Table 5 also shows the measured
38
M.I. Friswell
Table 4. The percentage changes in the natural frequencies for the damaged beam.
Modes
1 2 3 4 5
Element Stiffness Reduction 4.18 0.07 1.24 2.99 2.37
Beam Discrete Sinha et Spring al. (2002) 4.18 4.18 0.04 0.08 1.24 1.23 3.08 2.98 2.45 2.37
Plate Lee and Chung (2001) 4.18 0.04 1.20 2.99 2.34
4.18 0.04 1.22 3.07 2.69
natural frequencies for these crack depths. The damaged cantilever beam is modelled using the beam methods described earlier. The depth of the crack is optimised so that the percentage change in the first mode matches the experimental result, to allow for possible errors in measuring the crack depth. Tables 6 and 7 show the measured and predicted frequency changes for the 2 crack depths. The results clearly show that the differences in the natural frequencies predicted by the models are smaller than the measurement errors. Thus the simple models for cracked beams may be used with confidence in health monitoring applications. Table 5. The natural frequencies (in Hz) for the experimental cantilever beam example.
Modes
1 2 3
FE Model Undamaged 185.1 1159.9 3247.6
Undamaged 185.2 1160.6 3259.1
Experimental 2 mm Crack 6 mm Crack 184.0 174.7 1160.0 1155.3 3245.0 3134.8
Table 6. The percentage changes in the natural frequencies for the damaged beam with a 2 mm crack.
Modes
3.2
1 2 3
Element Stiffness Reduction 0.648 0.065 0.606
Discrete Spring 0.648 0.063 0.610
Sinha et al. (2002) 0.648 0.130 0.604
Lee and Chung (2001) 0.648 0.063 0.606
Experimental 0.648 0.052 0.433
Composite Structures
Composite structures have an excellent performance, although this deteriorates significantly with damage. Unfortunately damage, due to impact events for example, are difiicult to detect visually, and hence some method of non-destructive testing of these structures is required. Zou et al. (2000) reviewed the vibration based methods that are available to monitor composite structures. Since this paper considers inverse methods for
Damage Identification using Inverse Methods
39
Table 7. The percentage changes in the natural frequencies for the damaged beam with a 6 mm crack.
Modes
1 2 3
Element Stiffness Reduction 5.67 0.56 4.92
Discrete Spring 5.67 0.54 4.95
Sinha et al. (2002) 5.67 0.88 4.49
Lee and Chung (2001) 5.67 0.54 4.92
Experimental 5.67 0.46 3.81
damage estimation, this section will only consider the parameterisation of the damage in composite structures, and in particular the modelling of delaminations. Although composite structures have other modes of failure, such as matrix cracking, fibre breakage or fibre-matrix debonding (Ostachowicz and Krawczuk, 2001), these damage mechanisms produce similar changes in the vibration response to that obtained for damage in metallic structures. However delamination is a serious problem in composite structures, and has no parallel to damage mechanisms in other materials. Once the damage is parameterised then inverse methods, such as sensitivity analysis, may be applied. Zou et al. (2000) reviewed methods to model delaminations, and here we will concentrate on simple models. For example, if a structure is modelled with beam or plate elements, then only beam or plates elements should be used to model the structure with delaminations. Delamination occurs when adjacent plies in a laminated composite debond. For beam structures the simpliest case of a through width delamination, parallel to the beam surface, was modelled using four beam segments (Majumdar and Suryanarayan, 1988; Tracy and Pardoen, 1989). Separate beam elements were used above and below the delamination, and the constraints to join these elements to those of the undamaged parts of the beam needed to be applied carefully. Zou et al. (2000) detailed further development of these models. One difiiculty with using these models for parameter based identification is that changing the length and position of a delamination requires the model to be remeshed, and care must be exercised in calculating the associated sensitivity matrices. The techniques detailed by Sinha et al. (2002) for the position of cracks might be extended to this case. Paolozzi and Peroni (1990) highlighted that the most sensitive modes are those whose wavelength is approximately the same size as the delamination. Luo and Hanagud (1995) used a sensitivity based method to detect delaminations, and they also discovered that some modes split to give two closely spaced natural frequencies. 3.3
Joint Models and Generic Elements
One major difficulty in parametric approaches is that a model is required that accurately refiects the effect of damage on the mass and stiffness matrices. To some extent the situation is helped when low frequency vibration measurements are used because any local stiffness reduction will have a very similar effect on the dynamic response. Thus it is possible to use equivalent parameters, such as element stiffnesses, to model the damage. Generic elements (Gladwell and Ahmadian, 1995; Friswell et al., 2001) take this approach further by allowing changes to the eigenvalues and eigenvectors of the stiffness matrices
40
M.L Friswell
of structural elements or substructures. These changes are usually constrained so that properties such as the rigid body modes and the geometric symmetry are retained. Generic elements introduce flexibility into the joint in a controlled way. Other equivalent models, such as discrete rotational springs, offset parameters or changing element properties may also be used, although generic parameters do have advantages (Friswell et al., 2001). In particular, all models prejudge how the damage will affect the full model of the structure, whereas the generic element approach automatically finds the likely low frequency motion of the joint. Consider a two dimensional T joint constructed from three beam elements. Each node has three degrees of freedom and, since the substructure has four nodes, the substructure stiffness matrix has three rigid body eigenvectors and nine flexible eigenvectors (Titurus et al., 2003a). The lower eigenvectors have much simpler deformation shapes that are more likely to represent the motion the substructure would undergo in many of the global modes of the structure. Thus reducing the eigenvalues corresponding to these eigenvectors makes the joint substructure more flexible in the frequency range of the global dynamics, and may be used to model damage. Higher frequency eigenvectors of the substructure may also be included if the motion of the joint is more complex, however the lower eigenvectors of the joint are likely to adequately characterise the low frequency dynamics of the structure. Generic elements have been developed for use in model updating and may be considered as equivalent models of elements or substructures (Gladwell and Ahmadian, 1995). Law et al. (2001) applied generic elements to the finite element model updating of the Tsing Ma bridge in Hong Kong. Wang et al. (1999) used generic elements in damage detection, dealing with the simulated problem of damage detection in a frame structure with flexible L-shaped and T-shaped structural joints. The form of generic element parameterisation assumes that the damage only influences the stiffness properties and that the mass properties are modelled correctly. Thus only changes in the stiffness matrices are allowed. The eigenvalue problem for any selected sub-structure or element stiffness matrix can be written as (K^U^ - A,I) 0, = 0,
(^SUB>)T j^sUB^SUB ^
0 0
0 As
(3.6)
where *SUB ^ [ < ^ i , . . . , 0 „ ^ , ^ „ ^ ^ l , . . . , 0 „ ^ ^ J = [ * ^ , * 5 ] e SftnsuB^nsus^
(3.7)
and riR < 6. K^^^ is a sub-structure stiffness matrix, ^ ^ ^ ^ is the eigenvector matrix of K^^^, \i and (j)i are the ith eigenvalue and eigenvector of matrix K^^^, respectively. Sub-matrix As is a diagonal matrix of non-zero eigenvalues of matrix K^^^. The dimensions of these matrices depend on the size of the chosen sub-structure, where nsuB is a number of degrees of freedom of substructure and TIR < 6 is the number of rigid body modes, *i?, $ 5 are sub-matrices of * ^ ^ ^ corresponding to the rigid and structural modes, respectively. A modified set of sub-structure eigenvectors may be obtained by a linear transformation, as I^OR, ^os] = [*i?, * 6
SR
0
SRS
(3.8)
Damage Identification using Inverse Methods
41
where the index 0 denotes the original quantities and matrices without index 0 represent modified quantities. Notice that in Equation (3.8) the modified rigid body modes do not contain any of the structural modes. By rearranging Equation (3.6) and using Equation (3.8), the modified sub-structure stiffness matrix may be written as 1^1,1
K^™ ^ ^^^slAsSs^L
"'
^l,{nsuB-nR)
= ^^os
*5sblM
^{nsuB-nR),{nsuB-nR)
(3.9)
J
Equation (3.9) is the basis for generic element parameterisation for damage detection, /^i^i,... 1 K,{nsuB-nR),{nsuB-nR) ^^^ ^^c most general parameters for this parameterisation. Employing additional assumptions related to the geometric symmetry or antisymmetry of the corresponding eigenvectors will significantly reduce the total number of parameters. The sensitivity of natural frequencies with respect to these parameters is (
— = 6^ — dxj ^ dxj
Np
\
dKr (xr) (pi dxj
(3.10)
where Np is a number of parameterised substructures or elements, x/ is a group of parameters corresponding to /th substructure or element with corresponding stiffness matrix K/, x is a vector of all parameters, KQ is non-parameterised part of the global stiffness matrix, A^ is the ith eigenvalue and Xj is j t h parameter of a chosen parameterisation that is associated with the rth substructure or element. Consider a two dimensional T joint constructed from three beam elements. Each node has three degrees of freedom and, since the substructure has four nodes, the substructure stiffness matrix has three rigid body eigenvectors and nine fiexible eigenvectors. Figure 8 shows the nine flexible eigenvectors for this substructure, where the circles and dots represent the nodes and the dotted line is the undeformed joint. The finite element shape functions have been used to produce smooth deformation shapes. The lower eigenvectors have much simpler deformation shapes that are more likely to represent the motion the substructure would undergo in many of the global modes of the structure. Thus reducing the eigenvalues corresponding to these eigenvectors makes the joint substructure more fiexible in the frequency range of the global dynamics. Higher frequency eigenvectors of the substructure may also be included if the motion of the joint is more complex, how^ever the first two eigenvectors of the T joint were found to characterise the dynamics of the frame structure considered later. Gladwell and Ahmadian (1995) gave further explanation of the physical meaning of generic elements. 3.4
Distributed Damage
Teughels et al. (2002) presented a sensitivity-based finite element updating method for damage assessment that minimised differences between the experimental and predicted modal data. The parameterisation of the damage (both localisation and quantification) was represented by a reduction factor of the element bending stiffness. The number of unknown variables was reduced to obtain a physically meaningful result, by using a set of
42
M.I. Friswell
Figure 8. Substructure eigenvectors for a T joint.
damage functions to determine the spatial bending stiffness distribution. The updating parameters were then the multipHcation factors of the damage functions. The procedure was illustrated on a reinforced concrete beam and on a highway bridge (Teughels and Roeck, 2004).
4 An Example of Subset Selection using Generic Elements The proposed strategy is evaluated on a structure consisting of four thin-walled tubes connected to each other by four fillet welds. These joints were intentionally manipulated to produce one healthy and six damage cases. Titurus et al. (2003a) gave a detailed discussion of the identification results for the healthy/undamaged structure and Titurus et al. (2003b) described the estimation of the damage cases. Figure 9 shows the experimental structure, and Figure 10 shows the discretisation and experimental (EMA) measurement locations (the response was measured at the FE nodes). The finite element (FEM) nodes were placed at the measurement locations. Thus 32 degrees of freedom were measured, whereas the FE model contained 96 degrees of freedom (three degrees of freedom per node). The in-plane dynamics of the structure were measured, and the structure was supported in the free-free condition by elastic bands.
43
Damage Identification using Inverse Methods
Weldno.4
/
Weldno.3
A
290 D 40x20x2
Weldno.2
Weld no. 1
\
^^
z
D 60x20x2
1100
F i g u r e 9. The outline of the H-frame structure (dimensions in mm).
32 27
Sensor no.2 Sensorno.l
1
2
12
Figure 10. The discretisation of the H-frame structure.
4.1
D a m a g e cases
Figure 11 gives a detailed description of all of the damage cases. These cases were produced by the intentional incompleteness of one or more of the fillet welds used to interconnect the four tubular parts of the structure. The shaded lines show the welds completed for each of the four joints, for each damage state. Note that State VII has all welds in place and hence is the undamaged structure. A distinctive feature of this structure is its geometrical symmetry, which is likely to cause problems for damage location based on measured natural frequencies alone. Two different approaches will be evaluated; the first assumes that the influence of the transducer mass will be suflBcient to break the symmetry, whilst the second uses the measured mode shapes and their associated sensitivities. However, partial damage location will be tried, based on the use of natural frequencies alone. These damage cases were selected to give a reasonable coverage of all possible combinations of damage cases, within practical constraints. This section concentrates on the
44
M.L Friswell
single location damage case, and so only State VII, State VI and State V from Figure 11 will be studied in detail.
Figure 11. An overview of the damage cases considered.
4.2
Identification results for the damage cases
A full modal test was performed for each of the damage cases shown in Figure 11, however as the number of results is large only a selection will be considered here. The measurements were performed in the frequency range from 0 to 625 Hz. Table 8 gives the first nine measured natural frequencies for all of the damage cases. The fifth and sixth modes swap order between damage states III and IV. The last column corresponds to the undamaged/healthy structure, that is the structure with fully welded joints. Generally, the natural frequencies decrease with increasing level of damage, as a result of the decreasing stiff'ness of the structure. However, some small increases were observed in some natural frequencies from one case to another. One possible reason might be a small decrease in mass due to the absence of some weld material. Alternatively, taking the structure from the free-free suspension to undertake the welding may give small frequency changes due to slightly different suspension conditions. Table 8. The natural frequencies (Hz) of the healthy (State VII) and damaged (State I to VI) structure. Note that modes 5 and 6 swap between States HI and IV due to the damage. Mode
i
2 3 4 5 6 7 8 9
State I 27.63 118.64 126.38 169.77 264.77 279.64 298.91 393.61 550.55
State II 33.57 120.94 129.92 172.28 275.73 280.38 312.58 396.78 551.78
State III 34.18 120.74 130.64 172.32 275.42 280.24 317.15 399.06 552.71
State IV 48.60 125.04 138.97 174.77 280.09 300.44 342.10 418.68 560.06
State V 50.26 124.82 139.63 175.42 280.33 301.72 347.97 420.05 560.63
State VI 60.06 126.60 147.86 175.86 280.81 319.60 359.55 436.24 565.93
State VII 60.57 126.53 147.05 175.89 280.76 320.56 360.70 437.72 566.52
Damage Identification using Inverse Methods
45
Figure 12 shows the modal assurance criteria (MAC) matrices between the reference mode shapes of the healthy structure and mode shapes corresponding to the damaged cases. It is clear that the fifth and sixth modes interact and swap over between damage states I and VII. Another interesting feature shown by the MAC matrices is the relative insensitivity of the mode shapes to increasing damage, despite large changes in the natural frequencies. Titurus et al. (2003a) gave other experimental results, in particular the mode shapes corresponding to the healthy structure and further discussion of modelling issues.
Figure 12. MAC criterion between State VII and damage State I. The rows of each MAC matrix correspond to mode shapes of State VII while columns correspond to mode shapes of State I.
4.3
Parameterisation overview
Section 3.3 provided a detailed explanation of parameterisations to be used for damage location, however, for the sake of completeness, a summary is provided here. Parameterisation A is expressed in terms of two groups of generic elements. The first group consists of one generic substructure that models the parts of the structure containing the fillet welds, and two parameters are required for each substructure, as shown in Figure 13. The other group consists of three different generic elements, each requiring one parameter, as shown in Figure 13. Thus, parameterisation A requires the parameter vector x given by y:=[xi,X2,xs,X4,xs]
= [i^n, i^l2^ i4i^ ^41^'^ii]
(4-1)
where /^^^ denotes the (j, k) element of the matrix of the ith element/substructure, based on generic elements as detailed in Equation (3.9). The values of these parameters may be determined by model updating. This parameterisation allows partial localisation to the type of region where damage has occurred. Parameterisation B allows similar elements or substructures to have independent values of the corresponding generic parameters, to enable complete damage localisation. Parameterisation B requires 28 parameters for the H-shaped structure, as shown in Fig-
46
M.I. Friswell
parameter: KSI, K^22 parameter: K^H parameter: K^l parameter: K^^H Figure 13. Parameterisation A of the baseline model of the thin-walled H structure.
ure 14, and these are defined as, X2i—l
— f^iii
X2i — f^22i
x^+4 = /^ii,
^ ~
-'•5'^5'^5 4 ,
(4.2)
j = 5,6,...,24.
parameter: KH, K2 parameter: KH parameter: KH parameter: KH P9 PlO PhP2
Pll Pl2 Pis PhP4
Pl4 Pl5
Figure 14. Parameterisation B of the baseline model of the thin-walled H structure.
4.4
Partial damage location - natural frequencies alone
In this section, parameterisation A will be used for partial damage localisation, using only the measured natural frequencies. This simplified form of damage localisation is chosen as a first step in the damage detection of a geometrically symmetric structure. The first seven natural frequencies corresponding to the healthy and damaged structures, as well as the sensitivity matrix S, determined at the updated parameters values (Titurus et al., 2003a), were used to test the subset selection approach proposed in Section 2.2.
Damage Identification using Inverse Methods
47
Both the sensitivity matrix and the measured frequency differences for the considered damage cases, were normahsed by the corresponding measured natual frequencies. Since single-location damage states are of primary interest, State VI and State V will be used, as they represent two levels of damage in one fillet weld. State IV and State II will also be considered, as these are multi-location damage states with different levels of damage (see Figure 11). The results of damage location, in the form of subspace angles, are shown in Figure 15. Individual groups of columns correspond to particular model parameters. Within each group, corresponding to each parameter, the columns represent different comparisons of the damage cases (State II, IV, V, VI) with the healthy structure (State VII). Figure 15 suggests that the damage corresponds to parameter xi = KJI, which corresponds to the generic substructures containing the welded joints. Thus the damage is correctly localised to the welded joint. An important feature of this study is that even relatively small damage corresponding to State VI is readily observable and clearly identifiable. An increasing level of damage, represented here by State V, leads to improved and clearer identification of damage location or damage type.
Figure 15. Subspace angles for partial localisation, parameterisation A. The results disagree with the expectation that the increasing level of damage should lead to increasing subspace angles and consequently to a deteriorating quality of damage localisation, due to the increasing error in the linearisation of the original non-linear problem. However the level of damage for State VI is relatively small (the maximum difference between State VII and State VI is —0.83% for the first natural frequency, see Table 8) and therefore susceptible to measurement error. State V is characterised by a larger extent of damage, and therefore large differences in the natural frequencies (the maximum difference for this combination is —17.02% for the first natural frequency), produces smaller subspace angles. Figure 15 also gives the subspace angles for State II and State IV, and the angles corresponding to parameter 1 are smaller still. Although these are multi-damage cases, the damage still lies in the joints. Another noticeable and beneficial feature of the results is the insensitivity of the subspace angles from other parameters due to damage in the welded joints, reducing the possibility of false alarms. Figure 16 provides additional information in terms of a selection tree. A selection tree is a representation of the forward parameter subset selection where each node of the
48
M.L Friswell
tree corresponds to a selected subset and its colour represents the numerical value of the residual. The root of the tree represents the initial system, Equation (4.2). The branching factor and the depth of the tree are decided in advance and in our case results in binary selection trees with three levels corresponding to the selected parameter subsets. All three figures are determined using the first seven natural frequencies. The second best single parameter would be X5, a parameter that also effectively monitors the stiffness in the regions connected with welded joints. The important relative indicators of damage level for a given situation are the absolute values of the residuals provided by the amplitude bar on the left of the Figure 16, as the ability to reproduce the measurement vector decreases with increasing damage level and consequently the magnitude of residuals also increases.
Figure 16. Binary tree representing forward selection. State VII vs. State VI.
4.5
Complete damage detection - natural frequencies and mode shapes
The only way to deal with damage localisation for geometrically symmetric structures is to use spatial information in the form of mode shapes. Once again the analysis was limited to the single location damage case, i.e. State VII (healthy), State VI (level 1 damage at welded joint 4, see Figure 9) and State V (level 2 damage at welded joint 4). The subspace angles corresponding to the individual parameters (parameterisation B) were computed by the techniques presented in Section 2.2. However, since the vector 5z and the sensitivity matrix S contain elements corresponding to both the natural frequencies and the mode shapes, additional weighting must be included to represent the relative importance of the natural frequency and mode shape information, as presented in Section 2.2. There are problems in using mode shape information, particularly since the accuracy of their estimation from measured data is worse than for natural frequencies. This is
Damage Identification using Inverse Methods
49
compounded since the proposed approach uses the differences between the measured damaged and measured undamaged mode shapes. Thus only mode shapes that are sensitive to the candidate damage sites should be chosen. Table 8 shows the changes in the natural frequencies for the different damage states, and gives a good indication of this sensitivity. However the table shows the sensitivity of the natural frequencies, which is not necessarily the same as the sensitivity of the mode shapes. Certainly, if the natural frequencies change very little with damage, then the corresponding mode shapes will not be sensitive. Thus, of the first seven modes, modes 2, 3, 4 and 5 are unlikely to give useful spatial information (note that mode 3 has been excluded because of the slight increase in the natural frequency in State VI). The sensitivity of the mode shapes to damage also increases with mode number, as the mode shapes corresponding to higher frequencies, contain more local deformation. Mode 1 is a global mode and therefore its shape is insensitive to damage. The proposed approach using mode shapes will be demonstrated using spatial information from mode 6. Relative errors in the first seven natural frequencies and the difference in the mode shape elements were used. No further weighting was included, as similar results were obtained with other weighting values. Figure 17 shows the subspace angles corresponding to damage State V, and provides the correct indication of damage location, corresponding to parameter xj. This parameter belongs to the fourth generic T substructure, representing fillet weld number 4 (see Figure 14). Other significant parameters indicated by the subspace angles are parameters X2^ to X28, which are located on the crossbar neighbouring the damaged region. However the results from the frequency only estimation clearly indicated that the damage is located in the joints. Thus damage in welded joint 4 may be confidently predicted.
Figure 17. Subspace angles corresponding to State V, using seven natural frequencies and mode shape 6.
5 Methods using Mode Shapes This section considers a different approach where changes in the measured modes shapes are used directly to detect and locate damage. Farrar and Jauregui (1998a,b) compared several of these methods, such as the damage index method (Stubbs et al., 1992), mode
50
M.I. Friswell
shape curvatures (Pandey et al., 1991), the change in flexibihties (Pandey and Biswas, 1994) or the change in stiffness (Zimmerman and Kaouk, 1994). The example used was a road bridge with a concrete deck and steel supports. Different levels of damage were introduced, but the damage was only clearly located with most methods at the most severe level where the first natural frequency changed by over 7%, and the mode shapes changed significantly. The damage index method (basically a measure of changes in modal strain energy) was found to be the most promising. Other methods based on pattern recognition, often using neural networks, are also popular (Sohn et al., 2001, 2002; Trendafilova and Heylen, 2003). These methods essentially provide curve fits using interpolation functions and are not based on physical models. The lack of a physical model also limits the scope for damage prognosis. One of the major problems with methods using mode shapes is the incompleteness of the mode shapes. The number of measured degrees of freedom is always far smaller that the number of analytical degrees of freedom. Thus the mode shapes must be expanded or the analytical model must be reduced. In either case errors will be introduced because the model without damage is generally used for this reduction or expansion. The other source of incompleteness is the measured frequency range, which means that only a small number of modes will be measured. If damage causes a change in the stiffness matrix, this is further complicated because the high frequency modes have most effect on the elements of the stiffness matrix, but the lower frequency modes are generally measured. A huge number of different methods have been proposed and only a selection of the available methods will now be described. 5.1
COMAC
Perhaps the simplest method is the Coordinate Modal Assurance Criterion (COMAC). The usual Modal Assurance Criterion (MAC) correlates modes shapes by summing over the measured degrees of freedom. The COMAC sums over the modes and thus gives information about the correlation of the degrees of freedom (Lieven, 1988). If (puij is the j t h element of the zth mode shape vector for the undamaged structure, and (t)dij is the corresponding quantity for the damaged structure, then the COMAC for degree of freedom j is IILi4>uij(pui,j\\
Wl^i(pdij(pdij\\
Damage location is determined by those degrees of freedom with a low correlation between the healthy and damaged states. Note that local damage will affect all of the degrees of freedom to some extent, and so the changes in the mode shapes will not necessarily be local. 5.2
Balancing the Eigenvalue Equation
Suppose that the first r natural frequencies and mode shapes are measured. If A^^ are the eigenvalues and (j)di the mode shapes of the damaged structure, then [X^Md + Xdi^d + Kd](t>di = 0,
i = 1 , . . . , r,
(5.2)
Damage Identification using Inverse Methods
51
where M^, D^ and K^ are the mass, damping and stiffness matrices of the damaged structure, which are of course unknown. Suppose that damage only affects the stiffness matrix, then M^ = M^, D^ = D-^ and K^ = K^ — SK where M-u, Du ^-nd K^ are the mass, damping and stiffness matrices of the undamaged structure, which are assumed to be known. In practice model updating may be used to ensure that the model of the undamaged structure accurately represents the measured dynamics. SK is the unknown change in the stiffness matrix due to damage. Equation (5.2) then becomes, Li = [XdiMu + XdiBu 4- K^] (f)di = 5K(l)di,
i = 1 , . . . , r.
(5.3)
In Equation (5.3) the vector L^ on the left side of the equation may be calculated. If the change in the stiffness matrix due to damage is local then only those parts of the wStiffness matrix corresponding to the affected degrees of freedom will be non-zero, and hence the damage may be localised by the non-zero elements of the vector Lj. If more than one mode is measured then the contribution from each mode may be averaged using the root mean square for each degree of freedom. 5.3
Modal Strain Energy
Zhang et al. (1998) presented a method which compared the modal strain energy within the elements. For element j , with corresponding element stiffness K j , the element modal strain energy ratio for the iih mode (SERij) is
0.' K0i
CO-
where (j)i is the ith mass normalised mode shape, Ui is the ith natural frequency and K is the stiffness matrix of the structure. A damage indicator Pij is then defined as the difference of the element modal strain energy ratio before and after damage as,
d%
ut
where the subscripts u and d denote the healthy and damaged structures. 5.4
Direct and Minimum Rank Update Methods
Consider first the direct updating methods. The goal is often to reproduce the measured data (usually the modal model), by changing the stiffness matrix as little as possible (in some minimum norm sense). Historically these method were among the earliest in model updating (Priswell and Mottershead, 1995) and a number of generalisations have been proposed, depending on what is considered to be the reference quantities (Kenigsbuch and Halevi, 1998). A number of problems exist with the direct methods. There is no guarantee that the resulting matrices are positive definite (or semi-definite for structures with free-free modes), and extra modes may be introduced into the frequency range of interest. The standard methods do not enforce the connectivity of the structure, represented by the handedness of the matrices and the pattern of zero terms, although Kabe
52
M.I. Friswell
(1985) gave a method that enforced the expected connectivity. More fundamental is that forcing the model to reproduce the data does not allow for the errors that will be present in the measured data. Mode shapes, in particular, can only be measured with a limited accuracy. The major problem for damage location, and indeed for error location in model updating, is that all elements in the matrices may be changed. If only a small number of sites are modelled incorrectly (or are damaged) then only a small number of the matrix elements will be changed. Generally, because of the minimum norm optimisation in the updating method, all the matrix elements would be changed a little, rather than a small number of elements changed substantially. Thus the effect of any damage present would be spread over all the degrees of freedom making location difficult. Zimmerman and Kaouk (1994) and Kaouk and Zimmerman (1994) proposed that the change in the stiffness matrix should be low rank. This does not ensure that the change in stiffness will be local, as the stiffness change could be global but low rank. The method requires the rank of the stiffness change to be less than or equal to the number of measured modes used in the update. Zimmerman et al. (1995) gave an overview of this approach, and discussed issues such as the number of measured modes to use. Doebling (1996) extended the method by updating the elemental parameter vector rather than the global stiffness matrix. Abdalla et al. (1998, 2000) developed methods by minimising the change in the stiffness matrix, while enforcing constraints such as symmetry, sparsity and positive definiteness. The development begins by combining Equation (5.3) for all r measured modes to give, 5KVd = MuVdAl
+ BuVdAd + KuVd = B,
(5.6)
where V ^ = [^di (t>d2 --- (pdr] and A^ = diag [A^i Xd2 ^ - ^dr]
It may be proved (Zimmerman and Kaouk, 1994; Kaouk and Zimmerman, 1994) that the minimum rank of SK. is r, and that this minimum rank solution to Equation (5.6) is SK = B[B'^Vd]~^B'^. 5.5
(5.7)
Change in Flexibility
The flexibility matrix is the inverse of the stiffness matrix. In terms of the mode shapes the flexibility matrix C is
c =i=iEa;2\4>i4>J
(5.8)
'
where Aj and (pi are the ith natural frequency and mass normahsed mode shape and n is the number of degrees of freedom in the model. Note that the lower (measured modes) have the largest influence on the flexibility matrix. The flexibility method (Pandey and Biswas, 1994) compares the flexibihty matrices for the healthy and damaged structure, based on the r measured modes, as (5C = C„ - Cd
(5.9)
Damage Identification using Inverse Methods
53
where
A measure in terms of degrees of freedom is obtained by taking the maximum along each column of 5C
6 Sensor Validation The correct functioning of structural health monitoring systems requires that the sensors be functioning. Errors introduced by faulty sensors can cause undamaged areas to be identified as damaged. In many civil structures applications for health monitoring (such as bridges), ambient loads must be used for excitation. These loads are not known and may be measured or estimated as part of the health monitoring algorithm, which requires a large number of sensors. Sensor validation, where the sensors are confirmed to be functioning during operation, seems to have received little attention. The critical aspect in structural health monitoring is that there are usually more sensors than excited modes. This redundancy may be used, together with a modal model of the structure, to validate the sensor functionality. The control and chemical engineering community have considered the sensor validation problem, and have used models and sensor redundancy to good effect. However, these approaches usually use the faulty sensor to predict the response and look for errors between predictions and measurement. Clearly using the faulty sensor in the prediction process will propagate errors to the predicted responses. Often neural networks, or artificial intelligence approaches are used for the analysis. Friswell and Inman (1999) assumed that only the lower modes of the structure are usually excited, producing a large redundancy in the data. This has similarities to the principal component analysis used in chemical plant (Dunia et al., 1996; Dunia and Qin, 1998). Moreover, the approach seems to work only under the assumption of additive faults while giving erroneous results for the multiplicative faults case. Physically additive faults might arise from DC offsets in the electronic equipment and multiplicative faults might arise from calibration errors. The alternative used here, is to generate new residuals using the modal filtering approach which has similarities to the approach of Friswell and Inman. It is shown that these new residuals have interesting fault isolation properties. The approach is demonstrated on a subframe structure, although the method is completely general and may be applied to any structure for which a modal model is available. If necessary, such a model could be obtained from an identification experiment. For fault isolation a correlation index is proposed which is shown to correctly identify the faulty sensor. Faults may cause a variety of changes in the dynamic response of a sensor, and many of these are difficult to model. However the two most common faults, namely additive and multiplicative faults, are relatively straightforward to model. Physically additive faults might arise from DC offsets in the electronic equipment and multiplicative faults might arise from calibration errors. In this section the sensor faults are assumed to be additive and modelled as a constant signal added to the sensor response. The problem of
54
M.I. Friswell
detecting sensor faults is then transformed into the problem of the detection of the change in the mean of a Gaussian variable with known covariance matrix, which switches from zero under the no-fault condition to a mean value with unknown magnitude under the fault condition. This problem may be solved using a Ukehhood ratio test resulting in a X^ distributed variable which is then compared to a threshold. In order to decide which sensor or subset of sensors is most likely to be responsible for the fault, the so-called sensitivity tests are computed, which are also y^ distributed. 6.1
Sensor Validation Concepts
Although there is redundancy in the data, based on the number of sensors and the number of modes excited, it is still not straightforward to identify those sensors that are damaged. When all sensors are working it is possible to estimate the modal contributions to the response and therefore produce a predicted response that will give some idea of the accuracy of the model of structure and the extent of the measurement noise. However if a sensor is damaged, then using data from this sensor to estimate the modal participation factors will propagate the errors from the faulty channel through the estimate of the modal response to the estimate of the response in all channels. Thus to predict faulty sensors the sensors are split into two groups. If S represents the set of all sensors then these two groups are. Si — {sensors assumed to be faulty} Sw = {sensors assumed to be working}
(6.1)
Note that these two sets are disjoint so that SfnS^ = {},
SfUS^ = S.
(6.2)
Note that the distribution of faulty and working sensors seems to have been determined at the outset. In practice which sensors will be faulty is unknown and so every potential subset of faulty sensors must be tried. This approach has parallels with the subset selection technique in parameter estimation (Friswell et al., 1997; Millar, 1990). The difSculty in sensor validation, as in parameter estimation, is to determine which sensor or parameter subset is optimal. Note that for sensor validation, the number of assumed working sensors should be at least as great as the number of modes of interest. 6.2
Validation via Modal Filtering
Central to the proposed strategy for sensor validation is a modal model of the structure and also the estimation of the modal participation factors during operation. At any time instant, t^, the measured output is y {tk) =yk = H * q {tk) = H^rCLr (tk) + H^dCLd {tk)
(6.3)
where the modes have been split into those that are retained, ^^^ ^nd those that will be discarded, *d- If H-^; picks out those outputs that are assumed to be working (i.e. are elements of 5,^;), then we need to estimate qr,k = Qr (tk) from Yw.k = H^*rqr,fc + H^*dqd,fc
(6.4)
Damage Identification using Inverse Methods
55
where yw.k denotes the response at the fully functioning sensors at time tk and qd,fc = Qd {tk). Clearly the discarded modes in Equation (6.4) could be neglected and the pseudo inverse used to estimate cy^k from the resulting over-determined set of equations, as q^,fc = (H^*^)V^,/c
(6.5)
where {j denotes the usual Moore-Penrose pseudo inverse. This gives an estimate of the response at the functioning sensors as Yw.k = H ^ * r (fi-w^r)^
(6.6)
Yw.k = ^Yw^k-
There will be an error introduced because (H^*,)^H^$d^O
(6.7)
and a better estimate may be obtained by using the orthogonality of the modes as q,,fe = ( * j H ; ; ; M ^ , , H z . * r ) " ' ^InlM^^rYw.k
(6.8)
where M^^;^^ is the mass matrix reduced to the degrees of freedom corresponding to the functioning sensors in the set Sy^. Given that the mode shapes are assumed known, SEREP would be the most appropriate reduction method (O'Callahan et al., 1989). However, if the discarded modes lie outside the frequency range of interest then the estimator based on the pseudo inverse. Equation (6.5), will be adequate. The corresponding estimate of the response is Yw^k = H ^ * r ( * 7 H ^ M ^ , 3 i ^ ^ r ) ~ ^ ^JlilM^^rYw^k
= ^Yw^k-
(6.9)
Both approaches give a projector matrix P from the response space to the space of the lower modes. The quality of the model may be determined by reconstructing the response at the functioning sensors and producing the error as 5^,fc = ( I - P ) y ^ , f e .
(6.10)
Reconstructing the responses of the faulty sensors gives the error as ^/,fc = Yf,k - Hf^Ark
(6.11)
where H / picks out those outputs that are assumed to be faulty (i.e. are elements of Sf). In practice we do not know which sensors are working and which are faulty. Therefore the errors in Equations (6.10) and (6.11) are generated for all possible sets S^j and Sf. Of course the estimation of the modal participation factors has been performed at every time step, and so the errors will be produced at every time step. The average error over the time range of interest may be easily computed. The projector matrix, P , is constant for a particular choice of sets Sw and Sf and only needs to be computed once. Those sets where the error in the faulty sensor(s) is much greater than the error in the functioning sensors are then used to locate the faulty sensors.
56 6.3
M.L Friswell The Parity Space Approach
Abdelghani and Friswell (2001) introduced a different approach to treating the residuals, that performs better on systems with multiplicative sensor errors. Three residuals are required. The first is related to the modal residuals given above, and is essentially the negative of the residual in Equation (6.10), and is 7fc = [ H ^ * r (Rw^r)^
- l ] y^^,fe.
(6.12)
The second residual is similar, but the complete set of sensors (including any faulty sensor) is used to calculate the modal quantities. Thus 7^ = H ^ * ^ (*^)^ Yk - y^,fc.
(6.13)
The final residual is the difference between the two, namely a=7fe-7fe.
(6.14)
The damage correlation index is then given by
p^-^.21
(6.15)
where the expected value is over the time index k. This correlation index may be computed for each potentially faulty sensor, and hence any faulty sensor determined. 6.4
Example
The structure considered in this study consists of a suspended steel subframe used extensively in modal identification studies (Abdelghani et al., 1997). The structure was excited at two different locations using random noise inputs, and 28 accelerometers were used to measure the time response. The analysis was performed in the 0 — 500 Hz frequency range and 32000 data points per channel were collected at 1024 Hz sampling frequency. All 28 sensors were used to identify the experimental natural frequencies, damping ratios and mode shapes from the first 3000 data samples. The Balanced Realization algorithm using data correlations was used for the identification (Abdelghani et al., 1999). Only the 5 first modes were retained (up to 300 Hz) and the natural frequencies and damping ratios are given in Table 9. The corresponding real mode shapes were then used to generate the residuals. Faults are added to the measured signals to simulate realistic behaviour. For all cases the data samples 3000-4000 were used to generate the residuals. Suppose that an additive fault is simulated, where of 50% of the maximum response is added to sensor 6. Figure 18 shows the results using modal residuals and demonstrates that additive faults may be detected. Next a multiplicative fault to sensor 6 is introduced: the time responses are multiplied by a factor of 1.5. The modal residuals perform poorly on multiplicative faults. Figure 19
Damage Identification using Inverse Methods
57
Table 9. Frequencies and damping ratios identified using the Balanced Reahsation algorithm. Frequency (Hz) 60.72 156.32 190.66 229.19 287.11
Damping (%) 0.13 0.17 0.16 0.20 0.10
1.5
1
0.5
0
1
5
10 15 20 Sensor Number
25
Figure 18. Correlation coefficient ratio under faulty and non-faulty conditions using modal residuals: Sensor 6 has an additive fault.
gives the results using modal residuals and clearly the fault has not been detected. Figure 19 shows the results of the parity space approach and shows that the technique is able to clearly isolate multiplicative faults. Furthermore it seems from these experiments that the neighbouring correlation ratios are not influenced by the faulty sensor (s). Figure 21 shows the results for a complete loss of sensor 8. Only the ratio of the corresponding sensor changes, while the others stay relatively unchanged. Finally the correlation ratio seems to be related to the amount of damage introduced to the sensors. All of the above results are based on five modes and 28 sensor locations. As the number of modes used increases, the subspace in which the response hes increases, whereas the subspace in which the errors lie decreases. Thus the performance of the fault location scheme decreases. Similarly using more sensors improves the results, since by increasing the number of sensors the redundancy in the data is increased, which improves the detection and isolation of the faults. As an example a smaller number of sensors was used for the experiment, and more modes were included. Figure 22 shows the results using 10 sensors and nine modes, for the fault at sensor 8, and should be compared to Figure 21. Fault detection has now failed.
M.I. Friswell
58 1.5
1
0.5
0
1
5
10 15 20 Sensor Number
25
Figure 19. Correlation coefficient ratio under faulty and non-faulty conditions using modal residuals. Sensor 6 has a multiplicative fault.
1.5
1
0.5
0
1
5
10 15 20 Sensor Number
25
Figure 20. Correlation coefficient ratio under faulty and non-faulty conditions. Sensor 6 has a multiplicative fault.
Damage Identification using Inverse Methods
59
1.5
1
0.5
0
1
5
10 15 20 Sensor Number
25
Figure 2 1 . Correlation coefficient ratio under faulty and non-faulty conditions: complete loss of sensor 8.
1.5
1
0.5
0
1
2
5
8 12 13 16 Sensor Number
20
22
26
Figure 22. Correlation ratio under faulty and non-faulty conditions: Sensor 8 faulty. Nine modes used for the 10 sensors.
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M.I. Friswell
7 Conclusions This chapter has given a brief introduction to the huge Hterature available on the approaches of damage identification based on inverse methods. The sensitivity based methods to identify physical parameters using subset selection for error localisation has been suggested as a viable approach. However, many difficulties remain to be fully solved, such as the modelling error between the model and the physical structure, and the influence of environmental factors. The most promising route is to include measurements of temperature, humidity and other environmental variables within the model, although this requires more stringent conditions on modelling error. At the very least these errors give a lower bound on the level of damage that can be detected and localised, and this can be formalised using statistics from the response of the undamaged structure in its normal operating environment. One scenario is that damage location using low frequency vibration is undertaken to identify those areas where more detailed local inspection should be concentrated. The application of robust damage detection and location algorithms based on monitoring the in-service response of a structure remains a challenge, although the availability of a model does open the way to more accurate prognosis and the estimation of the remaining life.
8 Acknowledgements The author gratefully acknowledges the support of the Royal Society through a Royal Society-Wolfson Research Merit Award. The author also acknowledges the contribution of his co-workers, highlighted by reference to published papers throughout this chapter.
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K. Worden. Structural fault detection using a novelty measure. Journal of Sound and Vibration, 201:85-101, 1997. K. Worden, G. Manson, and N. R. J. Fieller. Damage detection using outlier analysis. Journal of Sound and Vibration, 229:647-667, 2000. L. M. Zhang, Q. Wu, and M. Link. A structural damage identification approach based on element modal strain energy. In Proceedings of ISMA23, pages 107-114, Leuven, Belgium, 1998. Q. W. Zhang, L. C. Fan, and W. C. Yuan. Traffic induced variability in dynamic properties of cable-stayed bridge. Earthquake Engineering & Structural Dynamics, 31: 2015-2021, 2002. D. C. Zimmerman and M. Kaouk. Structural damage detection using a minimum rank update theory. Journal of Vibration and Acoustics, 116:222-230, 1994. D. C. Zimmerman, M. Kaouk, and T. Simmermacher. On the role of engineering insight and judgement in structural damage detection. In Proceedings of the 13th International Modal Analysis Conference (IMAC), pages 414-420, Nashville, TN, 1995. Y. Zou, L. Tong, and G. P. Steven. Vibration-based model-dependent damage (delamination) identification and health monitoring for composite structures - a review. Journal of Sound and Vibration, 230:357-378, 2000.
Time-Domain Identification of Structural Systems from Input-Output Measurements Raimondo Betti* * Department of Civil Engineering and Engineering Mechanics, Columbia University, 640 S.W. Mudd Bldg., New York, NY 10027. E-mail:
[email protected] Abstract This paper presents a methodology that can be used for the identification of secondorder models of structural systems using dynamic measurements of the input and of the structural response. This approach (and its variations) starts from an identified first order model of a structural system and obtain estimation of the structure's mass, damping and stiffness matrices. For these approaches, both the full instrumentation option and the partial instrumentation option are presented. For the case of partial/limited instrumentation, five different model types have been considered showing, for each models, the limitations imposed on the identification by the lack of available data. Once these dynamic characteristics have been determined, structural damage can be assessed by comparing the undamaged and damaged estimation of such parameters. A damage representative parameter is introduced: this parameter uses the identified models to detect the location and amount of structural damage. This methodology has been tested on simulated numerical results and its effectiveness in determining structural damage is evaluated.
1 Introduction In engineering, the process of System Identification (SI) can be described as the identification of the properties of mathematical models that aspire to represent the dynamical behavior of systems using experimental data. The purpose and tools of system identification depend strongly on the applications for which the model is sought. In civil engineering, the goal of system identification is to obtain structural models of buildings and bridges that can be used for an accurate prediction of the structural response to future excitations as w^ell as for damage detection purposes. In mechanical and aerospace applications, system identification provides control engineers with an invaluable tool for the design of proper control systems. Because of such a broad range of applications, it is reasonable to expect that there will be models that will be more suitable for certain applications than for others. As an example, in civil engineering applications, a structure could be modelled either as a first-order state-space model, for the purpose of predicting its response to future excitations or of designing a vibration control system, or as a physically meaningful second-order mass, damping and stiffness model for damage assessment purposes. The identification of such models from the measured dynamic response has proven to be a tough challenge, and it is often referred to as the "inverse vibration problem". Some of the noteworthy efforts in the identification of linear structural systems include the works by Agbabian et al. (1991), Safak (1989a,b), Udwadia (1994), Beck and Katafygiotis (1998a,b), Lu§ et al.
68
R. Betti
(1999), Alvin and Park (1994); Alvin et al. (1995), Farhat and Hemez (1993), Doebling et al. (1993), Tseng et al. (1994) and DeAngelis et al. (2002). Although they have the common goal of identifying a mathematical/physical model of the system, all these works differ from the type of model they identify and from the methodology used in the identification process. In the time-domain, one model that is commonly used in mechanical and aerospace engineering applications is the so-called "first-order" model in which the dynamics of the system is represented by a set of matrices that are not based on physical laws. Such models with no physical bases are often derived from realization theory and that be considered as a "black box": their only requirement is that they closely map the input-output data without any consideration given to the physical laws. In control theory, a realization is a reasonable model to use when the only variables to control are measured variables. An important advantage of this type of identification is that it is capable of identifying complex models even with just few input-output measurements, provided that the excitation is "rich" in frequency so to excite all vibrational modes. Although it is quite popular in control-type applications, this type of representation of a system, characterized by non-physical parameters, is rather inconvenient for damage identification purposes. For damage assessment purposes, it is more convenient to work with models that have a physical foundation: among these, "second-order" models have the system's dynamics represented by a set of second-order differential equations whose coefficients are physical parameters, like mass, damping and stiffness. In structural applications, the most common tool is definitely represented by the Finite Element Method (FEM). However, upgrading FEM models of complex structures is quite cumbersome and requires solid experience by the engineer and substantial simplifications about the parameters to be identified. In addition, issues related to the ability of the FEM to uniquely identify the model parameters from data are still unresolved. A recent trend among the most successful methodologies for the identification of physical characteristics of a structure (e.g. mass, damping and stiffness) is to start from an identified first-order representation of the system and, through proper transformations, to obtain the physical structural parameters (Lu§ et al. (1999), Alvin and Park (1994); Alvin et al. (1995), Tseng et al. (1994) and DeAngelis et al. (2002)). The drawback of this type of approaches is that these methodologies are strongly limited by the available number of input-output measurements. In fact, the common point among these methodologies is the assumption of having a full set of instrumentation (a sensor and/or an actuator on each degree of freedom). Evidently, this type of instrumentation setup requires a strong financial and computational effort and, unfortunately, rarely happens in real life applications, where complex structures are usually instrumented with only few sensors. Not having a complete set of measurements (either input and/or output) impairs the complete identification of structural properties (Lu§ et al. (2003)) and only limited information on the structural stiffness, damping and mass distribution can be retrieved. The size of such matrices is related to the number of input/output dynamic measurements available. In this section, a new methodology for the identification of structural parameters such as the mass, damping and stiffness matrices of a system will be presented. This approach consists of 2 phases: 1) the identification of a first-order realization of the structural system using the input/output recorded time histories, and 2) the identification of a second-order models of the system, specifically mass, damping and stiffness matrix, by using a proper transformation from first-order to second-order model. The case of a full set versus an incomplete set of instrumentation will also be discussed.
Time-Domain Identification from Input-Output Measurements
69
2 Statement of the Problem Consider an N degree-of-freedom viscously damped linear structural system, subjected to r external excitations. The equations of motion for such a system can be expressed as: Mq{t) H- £q(t) + Kci{t) = Bn{t)
(2.1)
where q(t) indicates the vector of the generalized nodal displacements, with (*) and (") representing respectively thefirstand second order derivatives with respect to time. The vector u(t), of dimension r x l , is the input vector containing the r external excitations acting on the system, with B e SR^^'^ being the input matrix that relates the inputs to the DOFs. The matrices Ad G SR^^^, C e 5R^^^, and K e 5R^^^ are the symmetric positive definite mass, damping, and stiffness matrices, respectively. Let us assume that only m output time histories of the structural response are available, so that the measurement vector y(t), of dimensions m x 1, can be written as: y{t) = [{CMt)f iev q. Then eq.(3.1) can be approximated as • mxl
M„
r(9+l)Ur(q+l)xi
(3.5)
Time-Domain Identification from Input-Output Measurements
71
where Y=[y(0) y(l) y(2) ••• y(g) ••• y(/-i)] M = [ D BcT Bc*r Bc*''"^r ] and
u=
1" u(0) u(l) u(2) • • • u{q) ••• u(0) u(l) ••• u(g-l) u(0) ••• u{q-2)
[
u(0)
u(/-l) u(/ - 2) u(/ - 3)
•••
(3.6) (3.7)
(3.8)
n{l-q-l)_
The matrices M and U (previously defined in eqs.(3.3) and (3.4)) have been truncated according to the value of q. If the data is to have a realization, then the Markov parameters approximately satisfy the equation M = YU^ where U^ is the pseudo inverse of the input matrix U, and the error due to such an approximation decreases as q increases. However, for lightly damped structures, the integer g, and therefore the size of U, is practically too large to perform the inversion operation numerically. To improve the stability of the system, and thereby to make the problem better conditioned numerically, Phan et al. (1992, 1993) introduced an observer based identification concept (Observer KalmanfilterIDentification, OKID) in which onefirstidentifies an associated observer from which the system pulse responses are then recovered. To understand the role of this observer, let us rewrite the system input-output relations in terms of a new set of state space equations which are obtained by adding and subtracting the term Ry(fc) in eq. (2.4) as presented in Juang et al. (1993). This will lead to: x(fc + l)
= *x(fc) -h ru(Jk) + R((Cx(fc) + Du(fe)) - y(fc)) (* + RC)x(fc) + (r + RD)u(fc) - Ry(fc) *x(fc) + Tu{k) y(A:) = Cx(/c) H- Du(fc)
(3.9)
where ^ = (* + RC) f = [(r + RD) (-R)] u(fc) u{k) y{k)
(3.10) (3.11) (3.12)
The gain matrix R is chosen to make the system represented by eq.(3.9) as stable as desired. Although eqs.(2.4) and (3.9) are mathematically identical, eq.(3.9) can be considered as an observer equation and the Markov parameters of this new system, denoted as M, are called the observer's Markov parameters. If the matrix R is chosen in such a way that * is asymptotically stable, then C * r « 0 for /i > p, and we can solve for M from general input-output data using • mxl
'
^mx{{r-\-m)p-\-r)^{{r-\-m)p-\-r)xl
M = YVt
(3.13) (3.14)
72
R. Betti
where Y=[y(0)
y(l)
M = and
y(2)
c*^~'f
D cr c*r
u(0) u(l) 1/(0)
u(2) u{l) 1.(0)
y(/-i)]
y{p)
(3.16) u(/-l) v{l - 2) u{l - 3)
u{p) uip-1) u{p-2)
(3.15)
(3.17)
u{l-p-\)
1/(0)
It is important to note that p is now much smaller than q, and so the numerical difficulties are overcome. This development can also be viewed as the introduction of an observer to the system, with R being the observer gain. For further details on the formulation and on how to account for the initial conditions, the reader is referred to the work by Phan et al. (1991), Phan et al. (1992), Phan et al. (1993), Juang et al. (1993), Juang and Phan (1994a), and Phan et al. (1995). Having identified the observer Markov parameters, the true system's Markov parameters can be retrieved using the recursive formula: fc-i
Mfc = M^') + ^Mp)Mfc_i_i + M (2) D
(3.18)
i=0
where
M = [ M_i M, =
Mo
c*''r
[C(* + RC)'=(r + RD), M^^\
(2)1 .
Mk J .
Mp_i ]
Ml
(3.19)
- C ( * + RC)'=R]
fc=l,2,3,..
(3.20)
with M_i = D.
4 Identification of State-space Models from Markov Parameters Once the Markov parameters of the system have been determined, they can be used in building an Hankel matrix whose singular value decomposition leads to the identification of a state-space model of the system. This method is called the Eigensystem Realization Algorithm (ERA) and it is one of the most widely used and studied algorithms in the mechanical / aerospace engineering arena. Different formulations of the ERA algorithm include: 1) ERA in the Frequency Domain (Juang and Suzuki (1988)), 2) ERA in a recursive form (Longman and Juang (1989)), 3) ERA with Data Correlation (ERA/DC) (Juang et al. (1988)). Other ERA-based algorithms include the works by Yang and Yeh (1990), Juang and Phan (1994b), Phan et al. (1995), Moonen et al. (1989), and Lim (1998).
Time-Domain Identification from Input-Output Measurements
73
To briefly present the fundamental theory of ERA, let us consider an NDOF system, represented by afirst-orderdiscrete time state space model as in equation 2.4, and assume that r unit pulse tests have been performed on a system with m outputs; i.e. y(fc)= [ yi{k)
y2{k)
ym{k) ] .
Let us denote with y^(fc) a new vector, of dimension m, which represents the system's response at timefc(AT)to a unit pulse at time zero applied at input Uj (same as Markov parameters). In this way, we can regroup the data as
Y{k)=[yHk)
yHk)
y'-(fc) ] ,
k=l,2,...
(4.1)
and form the ms x rs Hankel data matrix
n'{k-i)
Y{k) Y(fc + 1)
=
LY(fc + s - l )
Y(fc + 1) Y(fc + 2)
Y{k + s-l) Y(fc + s)
Y{k + s)
Y(fc + 2 ( s - l ) )
(4.2)
where s is an arbitrary integer that determines the size of such a matrix. Looking at eqs. 2.6, it is possible to see that thefirstMarkov parameter, i.e. D, can be readily expressed as D = Y(0)
(4.3)
If the recorded data permits a realization, then the full data sequence can be generated from the triplet (C, *, T) via the following equation Y(fe) = C * ' ' " ^ ,
(4.4)
it = 1,2,3,..
which substituted into (4.2) leads to the following representation of the Hankel matrix
n'it)
i=0,l,....
cr
i+s-l
r cr+*r
(4.5)
Qpi+2(s-l)p
It can be shown that, if there exists a finite dimensional realization (C, * , T) of the data sequence Y(/c),fc=l,2,3,...,and if the dimension of a minimal realization is n, then rank 'Hf{i) = n,
^ s>n,
and i=0,l,2,...
(4.6)
Once the system's Markov parameters have been determined and the corresponding Hankel matrix has been built, let the singular value decomposition of ?t*(0) be denoted by
n'{o) = oe = usv^ = [ Ui U2 ]s0 00
= UiSVf
(4.7)
74
R. Betti
where Vmsxms and Vrsxrs are unitary matrices, and S is a square diagonal matrix (the nonzero partition of J^msxrs) whose dimensions are equal to the rank of the W{0) matrix. The basic theorem of the ERA realization states that, if the dimension of any minimal realization is n, then the following triplet is a minimal realization of the system for any s >n: S-2U/? 0 such that in [0,^]
ao 0, limAme = Am,
m = 1,2,... .
(2.18) (2.19)
Here, i?^(0,£) is the Hilbert space formed by the measurable functions / , / : (0,£) -^ R, such that both / and its first derivative / ' (in the sense of distributions) belong to the space L^(0,^) of the square summable real-valued functions on (0,^). One can notice that (2.15) can be read as a series Taylor expansion of the mth eigenvalue in terms of the variation 6g. In fact, in an abstract context, the integral term
142
A. Morassi
in (2.15) is the partial derivative of Xme with respect to the axial stiffness coefficient a^ evaluated, at e = 0, on the direction 6e. This partial derivative can be interpreted as the scalar product (in L^-sense) between the gradient ^ ^ ^ | e = o = (^m(^))^ ^^^ ^^^ direction b^, namely
The expression of the integral term in (2.15) shows that the so-called sensitivity of the mth eigenvalue to changes of the axial stiffness depends on the square of the first derivative of the corresponding mth vibration mode of the unperturbed system. When the perturbation be is localized in a small interval centered in XQ, XQ € (0,^), formula (2.15) indicates that the first order variation of the mth eigenvalue depends on the square of the longitudinal strain evaluated at XQ, see Section 3.2 for an analogous result in the extreme cases of cracks and notches modelled by translational elastic springs inserted at the damaged cross-sections. The explicit expression of the first derivative of an eigenvalue with respect to cracks or notches will be used in Sections 3.2 and 3.4 to identify localized damages in rods and beams by minimal frequency measurements. Analogous applications to discrete vibrating systems with a single localized damage are presented in Dilena and Morassi (2006). The analysis has hitherto been related to rods under axial vibration with free ends. However, it is clear that, under analogous assumptions, the asymptotic eigenvalue expansion formula (2.15) holds true for rods with different boundary conditions, such as, for example, supported {u{0) = 0 = u{£)) or cantilever {u{0) = 0, a[tju\tj = 0). 2.3
A Reconstruction Procedure
The linearized problem Let the free vibrations of the reference rod and the perturbed rod be governed by the eigenvalue problems (2.1), (2.4) and (2.13), (2.14), respectively. The coefficients a and p are assumed to satisfy conditions (2.2) and (2.3), respectively. In this section, the inverse problem of determining the perturbation b^ of the axial stiffness from measurements of the changes in the first M natural frequencies will be considered. The coefficient be is assumed to satisfy (2.7)-(2.9) and, in addition, the a priori condition SUpp be{x) = {xe{0,e)\be{x)^0}
C U
^
•
(2.21)
The above condition plays an important role in the present study. It should be noticed that there are situations important for applications in which (2.21) appears as a rather natural assumption. For example, if the reference beam is symmetrical with respect to X = i/2, then the eigenvalues Xme{bu)i Ame(^2e) corresponding to two perturbations bie{x), b2e{x) Symmetrical with respect to x = i/2, e.g. bu{fi — x) = b2e{x) in [0,^], and such that supp bu C (0,£/2), are exactly the same for every m = 1,2,... Loosely speaking, one can say that the Neumann spectrum cannot distinguish left from the right. To avoid the indeterminacy due to the structural symmetry, condition (2.21) will be assumed to hold. In practical diagnostic applications, (2.21) is equivalent to a priori know that the damage is located on an half of the rod, see, for example, Davini et al.
Damage Identification in Vibrating Beams
143
(1993) and Davini et al. (1995) for applications via variational methods. It should be mentioned that diagnostic techniques based on mode shape measurements (see Yuen (1985) and Rizos et al. (1990)), node measurements (Section 4), simultaneous use of resonance and antiresonances (Section 5.1) have been recently proposed to avoid the non-uniqueness of the damage location problem in symmetric beam structures. In order to illustrate the reconstruction procedure, the comparatively simple example of an initially uniform rod, with a = const, and p = const, in [0,£], is first considered. The eigenpairs of the reference rod are given by . , /~2~ mnx '^m{x) = d-jcos—j-^
,
a /m7r\2
^ ^ " " ~ [ ~ Y )
'
^ = 1'2,... .
,^ ^^. (2.22)
The rigid mode uo{x) is always insensitive to damage and, therefore, it will be omitted in the sequel. Putting the expressions of A^ and Um{x) for m > 1 into equation (2.15) gives
( fYi'^j^ \ ^ / 2 \
o mux
f
~T) \ e ) be{x)sm^-j-dx + r{e,m), where r{e,m) is a higher order term on e, see condition (2.16).
m = l,2,...,
(2.23)
The family {$m(x)}^=i, with $ ^ ( x ) = ^ ^ J ? ^ = ^ sin^ ^ is a basis for the square summable functions defined on (0,^/2). This means that any function / , with / : [0, £/2] —> R and / regular enough, can be expressed by the series oo
/W
= E
fm^rnix),
(2.24)
m=l
where fm is the mth generalized Fourier coefficient of the function / evaluated on the family {$„(a;)}-=i. By neglecting, as a first approximation, the higher order term r(e, m) in the asymptotic development of the mth eigenvalue and expressing b^ in terms of the functions {^m(x)}^=i,thatis oo be{x) = Y,Pek^k{x), fe=l
(2.25)
one has oo ^A^e = Y l ^rnkPek. fc=l
m = 1, 2, . . . ,
(2.26)
where a„e = ^ ^ ^ V ^ ' =
^rn^kdx=-^
rn = 1,2,...,
s m ^ - y - s m ^ - y , fc,m = l , 2 , . . . .
(2.27) (2.28)
A direct calculation shows that 2 3 Amk = j-Yp for ki^m, Amk = j-2-pforfc= m.
(2.29)
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A. Morassi
In real applications, only the eigenvalues of the first few vibrating modes are available. In fact, the number M typically ranges from 3 — 4 to 10. Therefore, rather than studying the solution of the infinite linear system (2.26), the following analysis will be focussed on its M-approximation^ that is the M x M linear system formed by M
6Xme = J2 ^"^kH^k.
rn^l,...,
M,
(2.30)
fc=i
where A^^ = Amk for k^m = 1,...,M, and {/9^}^^ are the coefficients of the Mapproximation of he{x) evaluated on the family {$^(x)}^ rm=iA direct calculation shows that / 1 \^ d e t A ^ , = (2M + l ) ( ^ ^ ,
(2.31)
(2.32) m,k = 1,...,M. Hence, the solution of (2.30) has the following explicit expression
^^'='^'''4^T\^^''-2II^^
S
^^^0' ^-1'-'^'
(2.33)
and, going back to (2.25), the first order stiffness variation is given by
Expressions (2.33), (2.34) clarify how the relative eigenvalue shifts influence the various Fourier coefficients of the stiffness variation b^{x). Assuming that the relative eigenvalue shifts are, in average, all of the same order, it can be deduced from (2.33) that for relatively large values of M (starting from M = 3 — 4, for example), the fcth Fourier coefficient l3^ is mainly influenced by the variation of the corresponding fcth eigenvalue. In fact, for a given k and, for example, for M = 4, the coefficient which multiplies SXke is equal to 0.78 about, whereas the coefficients of the remaining eigenvalue changes 6\j, j 7^ A;, take the lower value 0.22. This difference becomes significant as M increases. A n iterative procedure and a numerical algorithm The above analysis is based on a hnearization of Taylor's series expansion (2.23) for the eigenvalues of the perturbed rod. Therefore, the coefficient b^ found by (2.34) does not satisfy identically equations (2.23). The estimation of b^ can be improved by repeating the procedure shown above starting from the updated configuration a^^^ = a+^e? with be as calculated at the previous step.
Damage Identification in Vibrating Beams
145
This suggests the following iterative procedure for solving the invers^problem. The index e will be omitted in this part to simphfy the notation. Moreover, Am denotes the mth eigenvalue Xme of the perturbed rod. ITERATIVE PROCEDURE AND NUMERICAL ALGORITHM:
1) Let a^^^(x) = a(x), where a{x) is the axial stiffness of the reference rod. 2) For s = 0,1,2,...: a) solve the linear system ^ ^ ^ 7 ^ = E^™'^/^r''^ where {Xm\um)
m = l,...,M,
(2.35)
is the mth normalized eigenpair of the problem {a^'^uj + X^'^pu = 0
in (0, i),
(2.36) (2.37)
a^'\0)u\0) = 0 = a^'\()u\ti.
The numbers {(3^ Ifcli ^^^ ^^^ generalized Fourier coefficients of the unknown function h^^\x)^ b^^\x) = Y^k^i l3j^ ^^fc ? a-iid the matrix entries A^j^^^ are given by pi/2
AT=
Jo
^^M'^dx,
(2.38)
m,k = l,...,M.
b) Update the coefficient a{x): a^^+i) {x) = a(^) (x) + b^'^ (x)
in [0, i/2].
(2.39)
c) If the updated coefficient satisfies the condition
m=l \
^m
/
for a small given control parameter 7, then stop the iterations. Otherwise, go to step 2) and repeat the procedure. With the exception of simple cases corresponding to special stiffness coefficients, e.g. a{x) = const, in [0,^], the eigenvalue problem (2.36)-(2.37) does not admits closed form eigensolutions. Therefore, for the practical implementation of the identification algorithm resort to numerical analysis in order. The procedure herein adopted is based on a finite element model of the rod with uniform mesh and linear displacement shape functions. The stiffness and mass coefficients are approximated by step functions, that is a{x) = Ge = const.^ p{x) — pe ~ const. within the eth finite element. The local mass and stiffness matrices are given by Me = PeA ( J / ^ J / ^ ) ,
X e = a^h.'^
( \
'^
) ,
(2.41)
146
A. Morassi
where A is the element length. The discrete approximation of the eigenvalue problem (2.36)-(2.37) was solved by the Stodola-Vianello method. The derivative of the eigenfunctions was evaluated by using a finite difference scheme and the numerical integration was developed with a trapezoidal method. In solving the linear system (2.35), the determination of the inverse of the matrix A^l^'^ at each step 5, 5 = 1,2,..., is needed. If 5 = 0, then det A ^ f ^ = {2M-^l){4a^£)-^ by (2.31) and the inverse of the matrix A^jf^ exists. At the first step of the iteration scheme, 5 = 1, by (2.38) and recalhng that ulil = u'^^ + 5ume,
(2.42)
where Sume is a small perturbation term such that ||t5^Xme||/i^i -^ 0 as e —> 0, it turns out that
AT
= AT
+ SA^^.,
(2.43)
where SAmk,e —^ 0 as e —^ 0. Therefore, one can conclude that det A^l^^ = det A ^ f ^ + small terms as e -> 0,
(2.44)
and the inverse of the matrix A^j^ ^ is well defined. By proceeding step by step and within the assumption that the unknown stiffness coefficient is a perturbation of the initial one, the inverse of the matrix A^jf^ is well defined. If, during the iterative procedure, the coefiicient a^^"*"^^ violates the ellipticity condition (2.8), then the perturbation be is multiplied by a suitable step size a^^\ typically Q;(^) Z= 1/2, to obtain an updated coefficient satisfying (2.8) with ^o = i ^ ^^^xe[o,i] ci^^\x). This procedure is repeated at most five times during each step of the iterative process. After that, the iterations are stopped and the current stiffness distribution is taken as solution of the reconstruction procedure. Analogous considerations hold concerning the upper bound (2.8) with AQ = 2max^^[o,^] a^^^x). The small parameter of the convergence criterion (2.40) is taken as 7 = 1.0 • 10~^^ and an upper bound of 50 iterations was introduced. 2.4
Applications
The reconstruction procedure presented in the previous section is applied to identify stiffness variations caused by localized damages in longitudinally vibrating beams. The principal results of identification are summarized in the sequel. The experimental models are the two bars under free-free boundary conditions shown in Figure 1. Every specimen was damaged by saw-cutting the transversal cross-section. The width of each notch was approximately equal to 1.5 mm and, because of the small level of the excitation, during the dynamic tests each notch remains always open. In the first experiment, the steel rod of series HEIOOB (rod 1) shown in Figure 1(a) was considered, see Morassi (1997) for more details on dynamic testing. By using an impulsive dynamic technique, the first nine natural frequencies of the undamaged bar and of the bar under a series of two damage configurations {Dl and D2) were determined. The rod was suspended by two steel wire ropes to simulate free-free boundary
Damage Identification in Vibrating Beams
147
Notch (1.5 mm)
9
m Undamaged
Asymmetric Notch
Undamaged
Symmetric Notch
(a)
Damage D1
Damage D2
Damage 0 3
(b)
Figure 1. (a)-(b). Experimental models and damage configurations: (a) rod 1; (b) rod 2. Lengths in mm.
conditions. The excitation was introduced at one end by means of an impulse force hammer, while the axial response was measured by a piezoelectric accelerometer fixed at the centre of an end cross-section of the rod. Vibration signals were acquired by a dynamic analyzer HP35650 and then determined in the frequency domain to measure the relevant frequency response term (inertance). The well-separated vibration modes and the very small damping allowed identification of the natural frequencies by means of single mode technique. The damage configurations were obtained by introducing a notch of increasing depth at s = 1.125 m from one end. Table 1 compares the first nine experimental natural frequencies for the undamaged and damaged rod. The analytical model of the undamaged configuration generally fits well with the real case and the percentage errors are lower than 1% within the measured modes. The eigenfrequency shifts induced by the damage are relatively large with respect to the modelling errors and rod 1 provides an example for which the damage is rather severe from the beginning. The rod was discretized in 200 equally spaced finite elements and the identification procedure was applied by considering an increasing number of natural frequencies M, M = 1, ...,9. The chosen finite element mesh guarantees for the presence of negligible discretization errors during the identification process. Figures 2 and 3 show the identified stiffness coefficient when M = 3,5,7,9 natural frequencies are considered in identification, for damage Dl and D2, respectively. Convergence of the iterative process seems to be rather fast and, typically, less than 10 iterations are sufficient to reach the optimal solution. As it was expected from the representation formula (2.25), the reconstruction coefficient shows a wavy behavior around the reference (constant) value ao, see Figures 2 and 3. The maximum values of the positive increments are, in some cases, comparable with the maximum reduction in stiffness, which occurs near the actual damage location s = 1.125 m. However, the extent of the regions with positive change in stiffness becomes less important as the number of frequencies M increases and when more severe levels of
A. Morassi
148
1.5 x(m)
2.5
3 (a)
1.5 x(m)
1.5 x(m)
2.5
3 (C)
1.5 x(m)
2.5
3 (b)
(d)
F i g u r e 2. (a)-(d). Rod 1: identified axial stiffness EA for damage Dl with M=3 (a), M=5 (b), M=7 (c) and M=9 (d) frequencies. Actual damage location s=1.125 m.
Table 1. Experimental frequencies of rod 1 and analytical values for the undamaged configuration (the rigid body motion is omitted). Undamaged configuration: EA = 5.4454 X 10^ N, p = 20.4 kg/m, £ = 3.0 m; An% = 100 • {f^^dei _ fexpy^exp^ Damage scenarios Dl and D2; abscissa of the cracked cross-section s = 1.125 m; An% = 100 • (^fdam _ fundamyfundam Frequency values in Hz. Mode n 1 2 3 4 5 6 7 8 9
Undamaged Exper. Model An%
Damage Dl Exper. An%
Damag e D 2 Exper. A„%
861.4 1722.2 2582.9 3434.2 4353.6 5174.4 6020.0 6870.5 7726.4
805.7 1664.5 2541.9 3162.2 4332.2 4961.1 5750.2 6860.2 7302.3
737.6 1600.0 2505.3 3016.0 4310.2 4812.6 5616.0 6851.3 7095.8
861.1 1722.2 2583.3 3444.4 4305.5 5166.6 6027.7 6888.8 7749.9
0.00 0.00 0.02 0.30 -1.10 -0.15 0.13 0.27 0.30
-6.17 -3.35 -1.59 -7.92 -0.49 -4.12 -4.48 -0.15 -5.49
-14.37 -7.10 -3.00 -12.18 -1.00 -6.99 -6.71 -0.27 -8.16
149
Damage Identification in Vibrating Beams
0
0.5
1
0
0.5
1
1.5
2
2.1 2.5
1.5 X (m)
2
2.5
x(m)
3
0
0.5
1
3 (c)
0
0.5
1
(a)
1.5
2
2.5
3 (b)
1.5 X (m)
2
2.5
3 (d)
x(m)
10 8
lU 4
2 0
Figure 3. (a)-(d). Rod 1: identified axial stiffness EA for damage D2 with M=3 (a), M=5 (b), M=7 (c) and M=9 (d) frequencies. Actual damage location s=1.125 m.
damage are considered in the analysis. Prom Figures 2 and 3 it can been seen that the reconstructed coefficient can give an indication where the damage is located. The results of identification can be slightly improved by recalling that, from the physical point of view, the coefficient adam{x) clearly cannot be greater than the reference value ao{x). This suggests to a posteriori set the identified coefficient to be equal to a^^\x) wherever adam{x) > a^^^(x), see also Wu (1994). The results of most diagnostic techniques based on dynamic data strictly depend on the accuracy of the analytical model considered for the interpretation of the measurements and the severity of the damage to be identified. Rod 1 provides an example for which the analytical model (of the reference system) is very accurate and for which the damage is rather severe from the beginning. Therefore, in order to study the sensitivity of the proposed reconstruction procedure to small levels of damage, in the second experiment a steel rod of square solid cross-section with a small crack was considered (rod 2). By adopting an experimental technique similar to that used for rod 1, the undamaged bar and three damaged configurations obtained by introducing a notch of increasing depth at s = 1.000 m from one end, see Figure 1(b). The analytical model turns out to be extremely accurate with percentage errors less than those of the first experiment and lower then 0.2% within the first twenty vibrating modes, cf. Table 2. The percentage of frequency shifts caused by the damage are of order 0.1% and 0.3 —
A. Morassi
150 20 16
4 0 0
0.5
1
1.5 x(m)
2
2.5
0
0.5
1
1.5 x(m)
2
2.5
(b)
2016-
^12-
4-
0
0.5
1
1.5 x(m)
2
2.£
(C)
(d)
Figure 4. (a)-(d). Rod 2: identified axial stiffness EA for damage D3 with M=3 (a), M=5 (b), M=:7 (c) and M=9 (d) frequencies. Actual damage location s=1.000 m.
0.4% for damage Dl and D2, respectively. Therefore, for these two configurations it is expected that modelling errors could mask the changes induced by damage. This behavior is confirmed by the identification results, see Morassi (2001). Conversely, Figures 4,5 show that damage DS is clearly identified when the first 3 — 5 frequencies are measured. In this case, the results show a good stability of the identification when an increasing number of frequencies is considered in the analysis. Finally, the diagnostic technique was also tested on free-free longitudinally vibrating beams with multiple localized damages. We refer to Morassi (2007) for more details. 2.5
The Bending Vibration Case
In the previous sections, the problem of identifying the stiffness change induced by a damage in an axially vibrating beam from frequency measurements has been discussed. Here the corresponding problem for a beam in bending vibration is considered. The reconstruction procedure in the bending case The physical model, which will be investigate, is a simply supported Euler-BernouUi beam. The undamped free vibration of the undamaged beam are governed by the boundary value problem (jVy-Ap^ = 0 ^(0) = 0 = v{£),
in[0,^], (2.45)
Damage Identification in Vibrating Beams
151
Table 2. Experimental frequencies of rod 2 and analytical values for the undamaged configuration (the rigid body motion is omitted). Undamaged configuration: EA = 9.9491 X 10^ N, /9 = 3.735 kg/m, i = 2.925 m; An% = 100 • {f^odei _ jexpyjexp Damage scenarios D l , D2 and D3; abscissa of the cracked cross-section s = 1.000 m; An% = 100 • (/^"^ - fundamyjundam Prequcncy valucs in Hz. Mode n
Exper.
Undamaged Model An%
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
882.25 1764.6 2645.8 3530.3 4411.9 5293.9 6175.4 7056.7 7937.9 8819.9 9702.7 10583.8 11464.3 12345.2 13224.4 14104.0 14985.0
882.25 1764.5 2646.8 3529.0 4411.3 5293.5 6175.8 7058.0 7940.3 8822.5 9704.8 10587.0 11469.3 12351.5 13233.8 14116.0 14998.0
0.00 -0.01 0.04 -0.04 -0.01 -0.01 0.01 0.02 0.03 0.03 0.02 0.03 0.04 0.05 0.07 0.09 0.09
Damage Dl Exper. An% 881.5 1763.3 2644.0 3526.8 4408.8 5294.3 6168.8 7052.0 7937.5 8809.8 9697.3 10582.8 11449.0 12339.5 13222.8 14087.0 14979.0
-0.09 -0.07 -0.07 -0.10 -0.07 0.01 -0.11 -0.07 -0.01 -0.11 -0.06 -0.02 -0.13 -0.05 -0.01 -0.12 -0.04
Damage D2 Exper. A„%
Damage D3 Exper. An%
879.3 1759.0 2647.0 3516.5 4400.0 5295.3 6150.3 7039.5 7938.0 8782.0 9682.8 10581.3 11410.5 12331.5 13322.0 14039.0 14964.0
831.0 1679.5 2646.5 3306.0 4250.0 5287.8 5808.5 6864.3 7909.5 8340.0 9503.3 10514.8 10933.5 12158.0 13098.0 13543.0 14811.0
-0.33 -0.32 0.05 -0.39 -0.27 0.03 -0.41 -0.24 0.00 -0.43 -0.21 -0.02 -0.47 -0.11 +0.74 -0.46 -0.14
-5.81 -4.82 0.03 -6.35 -3.67 -0.12 -5.94 -2.73 -0.36 -5.44 -2.06 -0.65 -4.63 -1.52 -0.96 -3.98 -1.16
where v = v{x) is the transversal displacement of the beam, V^ is the associated natural frequency and p = p{x) denotes the linear mass density. The quantity j = j{x) = EJ{x) is the bending stiffness of the beam. E is the Young's modulus of the material and J = J{x) the moment of inertia of the cross-section. The function p is assumed to satisfy conditions (2.3). The bending stiffness j is such that jeC^{[OJ]),
j{x)>Jo>0
in[0,£],
(2.46)
where jo is a given constant. Under the above assumptions on the coefficients, problem (2.45) has an infinite sequence of eigenpairs {{vm^ \m)}m=i^ with 0 < Ai < A2 < ..., limm-^oo A^ = 00 and where the mth vibration mode is assumed to satisfy the normalization condition /^ pv'^ = 1 for every m, m > 1. In analogy with the axial case, it is assumed that a structural damage can be described within the classical one-dimensional theory of beams and that it reflects on a reduction of the effective bending stiffness, without introducing changes on the mass distribution. Following the analysis presented in Section 2.2, the bending stiffness of the damaged
152
A. Morassi
0
0.5
1
1.5 x(m)
2
2.5
0
0.5
1
1.5 x(m)
2
2.5
(C)
0
0.5
1
0
0.5
1
1.5 x(m)
2
2.5
1.5
2
2.5
x(m)
(b)
(d)
Figure 5. (a)-(d). Rod 2: identified axial stiffness EA for damage D3 with M = l l (a), M=:13 (b), M=15 (c) and M=17 (d) frequencies. Actual damage location s=1.000 m.
beam is taken as j^{x)
=j{x)-^be{x),
(2.47)
where the variation b^ is assumed to satisfy the following conditions: i) (regularity) (2.48)
b,€C\{0,e]); ii) (uniform lower and upper bound oi je) there exist a constant Jo such that jo<Je{x)<Jo
m[0,£];
(2.49)
iii) (smallness) ML^=eO{\\Jh2),
(2.50)
for a real positive number e. The free bending vibrations of the damaged beam are governed by the eigenvalue problem a